Formally, the classical continuum "exists" in the sense that that it is possible to axiomatically define connected and compact sets of dimensionless points that possesses a model that is unique up to isomorphism thanks to the categoricity of second order logic.

But the definition isn't constructive and is extensionally unintelligible for some of the reasons you pointed out in the OP. Notably, Dedekind didn't believe in the reality of cuts of the continuum at irrational numbers and only in the completeness of the uninterpreted formal definition of a cut. Furthermore, Weyl, Brouwer, Poincare and Peirce all objected to discrete conceptions of the continuum that attempted to derive continuity from discreteness. For those mathematicians and philosophers, the meaning of "continuum" cannot be represented by the modern definition that is in terms of connected and compact sets of dimensionless points. E.g, Peirce thought that there shouldn't be an upper bound on the number of points that a continuum can be said to divide into, whereas for Brouwer the continuum referred not to a set of ideal points, but to a linearly ordered set of potentially infinite but empirically meaningful choice sequences that can never be finished.

The classical continuum is unredeemable, in that weakening the definition of the reals to allow infinitesimals by removing the second-order least-upper bound principle, does not help if the underlying first-order logic remains classical, since it leads to the same paradoxes of continuity appearing at the level of infinitesimals, resulting in the need for infinitesimal infinitesimals and so on, ad infinitum.... whatever model of the axioms is chosen.
Alternatively, allowing points to have positions that are undecidable, resolves, or rather dissolves, the problem of 'gaps' existing between dimensionless points, in that it is no longer generally the case that points are either separated or not separated, meaning that most of the constructively valid cuts of the continuum occur at imprecise locations for which meta-mathematical extensional antimonies cannot be derived.
Nevertheless this constructively valid subset of the classical continuum remains extensionally uninterpretable, for when cut at any location with a decidable value, we still end up with a standard Dedekind Cut such as (-Inf,0) | [0,Inf) , in which all and only the real numbers less than 0 belong to the left fragment, and with all and only the real numbers equal or greater than 0 belonging to the right fragment, which illustrates that a decidable cut isn't located at any real valued position on the continuum. Ultimately it is this inability of the classical continuum to represent the location of a decidable cut, that is referred to when saying that the volume of a point has "Lebesgue measure zero". And so it is tempting to introduce infinitesimals so that points can have infinitesimal non-zero volume, with their associated cuts located infinitesimally close to the location of a real number.

The cheapest way to allow new locations for cuts is to axiomatize a new infinitesimal directly, that is defined to be non-zero but smaller in magnitude than every real number and whose square equals 0, as is done in smooth infinitesimal analysis, whose resulting continuum behaves much nicer than the classical continuum for purposes of analysis, even if the infinitesimal isn't extensionally meaningful. The resulting smooth continuum at least enforces that every function and its derivatives at every order is continuous, meaning that the continuum is geometrically much better behaved than the classical continuum that allows pathological functions on its domain that are discontinuous, as well as being geometrically better behaved than Brouwer's intuitionistic continuum that in any case is only supposed to be a model of temporal intuition rather than of spatial intuition, which only enforces functions to have uniform continuity.

The most straightforward way of getting an extensionally meaningful continuum such as a one dimensional line, is to define it directly in terms of a point-free topology, in an analogous manner to Dedekind's approach, but without demanding that it has enough cuts to be a model of the classical continuum. E.g, one can simply define a "line" as referring to a filter, so as to ensure that a line can never be divided an absolutely infinite number of times into lines of zero length, and conversely, one can define a collection of "points" as referring to an ideal, so as to ensure that a union of points can never be grown for an absolutely infinite amount of time into having a volume equaling that of the smallest line. This way, lines and points can be kept apart without either being definable in terms of the other, so that one never arrives at the antimonies you raised above.

Usually these sorts of discussions begin on the wrong foot by conflating communism with state capitalism under a ruling party, that is a situation resembling modern day corporate America in many respects, which is ironically reinforced by "communist" hating conservatives refusing to support progressive taxation.

I'm no Marx expert, but understand that he viewed communism descriptively as an inevitable outcome of capitalism, as much as he did as a moral imperative. With modern society's inevitable transition to universal income in the coming years, the appeal of communism seems besides the point.

Suppose y = sin(cos(x)). Which (sin or cos) would you say is inside, and which outside? — bongo fury

Quine was presumably referring to the stratification of types originally proposed by Russell, which ensures that the a map between type universes resides in a universe that is higher than both of the input and output universes. We might recall the fact that each universe contains a subclass that is isomorphic to the previous universe, as represented by the quotation marks in the liar sentence. So if we start with the highest level universe that we say contains everything we regard to be true, and use it to build in stepwise fashion an infinitely descending chain of so-called object languages that are each the meta-language of their predecessor, the liar sentence can be interpreted as stream of fluctuating truth values with respect to isomorphic, but non-identical terms of different types.

By contrast, both Sin and Cos are maps of type Real --> Real, i.e maps between terms of Reals, where the type Real --> Real resides in the same universe as the type Real, as does any function of functions of ... functions of reals; for they all reside in the universe definable in terms of second order logic. Following their example, we could alternatively interpret the liar sentence as directly referring to a stream of fluctuating values, where the stream and its values all exist in the same universe as a binary approximation to those trigonometric functions.

The identification of anything is subjective and relative to convention. E.g, we don't get hung up about whether or not today's chair is said to be same as tomorrows chair, so why should we treat persons as having objective haecceity ?

Tibetan monks might have their politico-cultural reasons for objecting to the Chinese government choosing the next Dalai Lama, but do they really have a metaphysical leg to stand on?

So the idea of persons as real and local spatial-temporal objects with objective physical boundaries is fundamentally incompatible with the idea that persons can be reincarnated. — sime


Beings are not only objects, they are also subjects of experience, and the nature of subjective experience is not necessarily describable in those terms. — Wayfarer

Are subjects of experience observable and identifiable or not?

Also consider the discovery of tulkus in Tibetan Buddhism. They are sought out by various means and subjected to examination and are said to be clearly discerned as incarnations of previously-existing figures. As already mentioned, Buddhist culture assumes the reality of rebirth as a matter of course, even despite the tension with the no-self principle. — Wayfarer

So Tulkas are observable subjects of experience who are interpretable in terms of types of objects?

In respect of the question of identity, Buddhists will respond, if you ask them, ‘are you the same person you were as a child?’ ‘No’. ‘Then are you a different person?’ Also, ‘no’. There is a continuity, but also change. I don’t think Buddhism has a difficulty with that. Overall, I find the Buddhist attitude congenial in these matters.

So I’m not really seeing your philosophical objection at this point. — Wayfarer

I'm not objecting to Buddhist sentiment to the extent that they understand that identity relations are arbitrary psycho-linguistic constructs that necessitate their semantic conclusions. In the case of the no-soul rebirth paradox, if concepts related to personhood aren't part of one's fundamental ontology, for example because one considers concepts of personhood to be unreal because one considers persons to be semantically reducible to impersonal forces of nature, then rebirth follows as a tautological conclusions, since the personhood concepts of life and death are both eliminated in the final analysis of of reality. In which case empirical evidence for rebirth is meaningless.

By contrast, if one conceives of persons as being fundamentally real and local token-objects whose existence is ontologically fundamental, then permanent death without reincarnation follows as a matter of tautology, and there cannot exist evidence to the contrary - for even allegedly successful past-life regressions must be discounted as illusory if one holds ones concept of persons as tokens as sacrosanct.


So the idea of persons as real and local spatial-temporal objects with objective physical boundaries is fundamentally incompatible with the idea that persons can be reincarnated. One of the concepts must give way to the other, and the question cannot be settled by appealing to empirical evidence, for the very meaning of "empirical evidence" lies downstream of this ontological decision.

You may not be familiar with the research. It wasn’t based on 'past-life regression'. The cases Stevenson sought out were those where children claimed to be someone other than who they were known to be e.g. would start saying 'your not my family' or 'this is not my home, I live in (some other place)' etc. Then the researchers would look for evidence of that claimed previous identity, trying to identify death notices, locations, and other details to corroborate the infant's story. — Wayfarer

Yes, I wasn't questioning the veracity of anecdotes such as the one you mention, rather I'm pointing out that to interpret such cases as being "evidence for reincarnation" is relative to a convention that defines personal identity in terms of memories, by which the person is said to be reincarnated. Which is why I do not take such cases with special seriousness - not because I am assuming that such reported cases cannot be happen as described, but because I consider the identity of persons to be arbitrary and decided by convention, and ultimately grounded in either psychological habits and prejudice or in the utility of adopting the chosen identity criteria.

For example, lets assume that the account you mention is accurate and defies mundane natural explanations. Then unless one has defined personhood in terms of personal memories, one cannot conclude that the child is a reincarnation of the previous person he is said to remember. In which case all that one concludes is that the child presently has abnormal access to novel information of historical significance.

Certainly, the child-as-token is not a previous person - by definition of "token". Compare this situation to a caterpillar-token that is said to become a butterfly-token. In that case, we don't insist that the butterfly remembers his life as a caterpillar in order for us to identify the caterpillar with the butterfly, rather we identify their tokens as being parts of a greater token on the basis of temporal continuity. Whereas in the case of the child, there is no apparent spatio-temporal continuity for us to say that the child was the becoming of the previous person, and instead we bridge their lives via a notion of "memory continuity", in spite of the fact that we rarely if ever employ such criteria in our own lives when we ordinarily identify ourselves and our loved ones over time.

As I mentioned to Philosophim, the point about the children with past-life recall is that there is at least the possibility of validating their statements against documentary and witness accounts, something which is obviously not possible with near-death experiences, as they are first-person by definition. — Wayfarer

Our memories are mutable. We continually create, delete and edit our memories in real time, including the memories that we interpret as being veridical. In general we don't interpret amnesia as constituting proof of personal absence during the past. So why should the possession of a veridical memory be interpreted as constituting proof of having witnessed the past? When it comes to conceptions of personal identity, why should ownership of memories be taken more seriously than ownership of a collection of disposable photographs?

The idea that studies of past life regression can verify or refute reincarnation, is in relation to a convention that defines personal identity in terms of memory possession, together with a block-universe conception of the past that memories are considered to refer to in a manner analogous to time travel. So I don't interpret studies of past life regression as drawing deeper metaphysical conclusions, regardless of whether such conclusions are positive or negative, than our pragmatic judgements of object identification.

That’s all well and good if your criteria of reincarnation is as slack as a good impression of that person or just imitation. Personhood has a more strict definition of what a person is as it covers what that person has experienced in life their memories made, habits personality traits and just general character. The issue boils down to personal identity and what it means to be you. — kindred

I'm arguing that even a supposedly strict definition of personhood is slack. Slackness is an inexorable feature of identity criteria; any application of identity criteria to any problem of philosophy leads to superficial and incomplete conclusions that are products of linguistic convention. At best, one's conclusions are circular and merely reiterate the identity criteria that one employed.

What conditions are required for the reincarnation of Elvis Presley?

In my pragmatic view, a good Karaoke singer who does a reasonable impersonation of Elvis on stage, can be said to be of roughly the same "type" as Elvis, at least until the end of the impersonation.

I don't consider the questions of reincarnation to run deeper than that, because identity criteria are inexorably vague, conflicting and decided by convention or psychological prejudices. So why should it be assumed that the question of reincarnation has a definite and absolute answer that transcends our conventions?

The main problem for me is, why can we read a→(b∧¬b) as "a implies a contradiction" but not ¬(a→(b∧¬b)) as "a does not imply a contradiction? — Lionino

In general, the consistency of an axiomatic system isn't provable in an absolute sense due to Godel's second incompleteness theorem; the upshot being that consistency is a structural property of the entire system that isn't represented as a theorem by the system if it is sufficiently powerful.

Suppose that the logic concerned is weaker than Peano arithmetic, such that it can prove its own consistency. Then in this case, a proof of ¬¬a metalogically implies that ¬a isn't provable, i.e that a does not imply a contradiction.

But if the axiomatic system contains Peano arithmetic such that the second incompleteness theorem holds, then a proof of ¬¬a does not necessarily imply the absence of a proof of ¬a, since Peano arithmetic cannot prove its own consistency.

Sure, but that's not really what the example is there to assert, as is clear from the rest of the paragraph. They mentioned replacing the fact about dogs 2+2 = 4 in the next line. It's "if a statement is true, then that statement is implied by any statement whatever," which is straightforwardly counter intuitive. — Count Timothy von Icarus

That's true of classical logic, and more specifically it's fragment known as intuitionistic logic, due to the fact that the respective rule of implication is essentially the logic of functions (including those that ignore their arguments to produce a constant value), rather than the logic of causality - which is described by relevance logic and linear logic.

As for "Modern Symbolic Logic", it doesn't have a well-defined meaning since it refers to a plurality of logics that are separately described in terms of different mathematical categories.

But consider the fact that the halting behaviour of two identical algorithms stands and falls together. So although there does not exist an infallible universal halting tester, there exists an infallible special-case halting tester for any given algorithm, namely a copy of that very algorithm.

Although an epistemic limitation falls short of a metaphysical proof, I am sympathetic to the idea of free will, because in my opinion the conceptual distinction between free will and determinism rests upon a belief in absolute infinity, which i reject.

In my view, to say that "A => B is necessary true" in the sense of material causation, is to say that there exists a Z such that "A => Z is necessarily true" and "Z => B is necessarily true". If we reject the idea that this definition can appeal to actually infinite recursion, then the use-meaning of " A => B is necessarily true" in any given context must eventually bottom out to a finite chain of implicative reasoning, in which the meaning of "necessarily true" is left undefined.

A simpler way of putting it, is to say that we make up the meaning of " A => B is necessarily true" as we go along. This proposition doesn't have precise a priori meaning, and so isn't contradicted by a future discovery that A => B fails to hold, rather the proposition meant by the sentence "A=> B is necessary true" changes on discovery that A => B fails to hold.

Bertrand Russell's Principia Mathematica was nominalist; he treated sets as merely a means of referring to groups of particulars, partly in response to Set theoretic paradoxes, but the approach made it impossible to describe all of mathematics.

This demonstrates an unconscious tendency of nominalism; why do nominalists have a tendency to appeal to an ontology based on the existence of particulars, as opposed to an ontology that starts from a united whole?

In Bertrand Russell's case, it was in the hope of making analysis tractable in piecemeal fashion, in contrast to the British Idealists who might also be described as nominalist, but who considered reality to consist of a single holistically unified entity. But this makes analytics impossible, since it implies that a local material change to reality causes the meaning and hence definitions of the rest of reality to change.

A related example is Godel's trick in his ontological proof of God as discussed in the other thread, which was to define a property P so as to enforce the condition

¬(g → P(g) ∧ g → ¬P(g))

i.e. ¬¬g, which is a classically acceptable proof of existence.

Just because a set S is declared to be "infinite", doesn't imply that S possesses a literally non-finite number of elements, rather it only implies that a bijection between {0,1,2..n} and S isn't or cannot be specified a priori and that n cannot be bounded a priori.

And even when a bijection between S and the set of natural numbers is specified, this only implies that S must be understood in terms of a monotonically increasing process rather than in terms of a completed basket of goods.

In both cases, the use-meaning of "infinity" should be understood to mean "finitization is decided by circumstances that are external to the specification logic".

So philosophical or practical questions about "infinite facts", as opposed to mathematical questions concerning the definition of mathematical infinity, should always be decided by elaborating assumptions until the facts concerned are "finitized". The presence of infinity in a non-mathematical question is only an indication that the question concerned isn't well-posed.

I don't see where that is implied in the argument.

P(ψ)≡¬N(ψ) — sime


If N is supposed to mean necessary existence, that is a rejection of axiom 5. — Lionino

N was supposed to mean the possibility modality (N standing for Negative Properties, in order to stand for the opposite of Positive Properties). The question here I was interested in, is how to give a syntactical definition of Positive Properties such that the resulting argument follows as a valid tautology in some modal logic. This was partly in order to help clarify the the definitions Godel provided, even his assumptions need to be altered slightly and the resulting argument and its conclusion aren't quite the same.

For example, taking Positive properties to refer to what is necessarily true of all individuals in every possible world, turns Axiom A2 into the definition of a functor, which is rather tempting. It also makes the possibility of god follow as a matter of tautology.

Also, Godel's definition of essences seems close to the definition of the Categorical Product. So why not take the essence of an individual to be the conjunction of his properties?

One thing I overlooked was that God was defined as referring to the exact set of positive properties, which would mean that according to my definition of P, all individuals would be identical. But then supposing we weaken the definition of "Godliness" to refer to a set that contains all the positive properties and possibly some of the negative (i.e contigent) ones?

I think there is quite a few pedagogically useful questions here.

It seems to me you are thoroughly confused — Lionino

about what?

I think the most remarkable and amusing part of Godel's argument, is in the beginning before the use of modal logic, in which he argues for the existence of a 'god term' by turning the principle of explosion on its head.

Constructively speaking, an existential proposition is proved by constructing a term that exemplifies the proposition, as per the Curry Howard Isomorphism. Classically speaking, an existential proposition can also be derived by proving that it's negation entails contradiction, as per the law of double negation.

In Godel's proof however, he defines a so-called Godliness predicate G, where as usual ~G(x) corresponds to the principle of explosion

G(x) --> B(x)
G(x) --> ~B(x)

where B is any predicate.

But in Godel's case, he defines G as only implying properties that satisfy a second-order predicate he calls "Positivity", which is a predicate decreeing that G(x) --> B(x) and G(x) --> ~B(x) cannot both be true.

So in effect, Godel crafted a non-constructive proof-by-absurdity that implies the existence of a god term on the basis that non-existence otherwise causes an explosion! this is in stark contrast to the normal constructive situation of proofs-by-absurdity in which a term exemplifying a negated existential proposition is constructed in terms of a function that sends counterexamples to explosions.


The rest of Godel's proof is unremarkable, since he defined G as implying it's own necessity, meaning that if G is said to be true in some world, then by definition it is said to be true of adjacent worlds, which under S5 automatically implies every world.

The irony of Modal Logic is that there are so many alternatives to choose from, corresponding to the fact that Logic and a forteriori modal logic, has no predictive value per se. But modal theologicians aren't using Modal Logic to derive or express predictions, rather they are using Modal Logic to construct a Kripke frame with theologically desired properties. So ontological arguments aren't necessarily invalid for achieving their psychological and theological purposes, provided they aren't construed as claims to knowledge.

In fact, i'm tempted to consider Anselm's argument to be both valid and sound a priori, and yet unsound a posteriori. This is due to the fact that although our minds readily distinguish reality from fiction, I don't think that this distinction is derivable from a priori thought experiments.

S5 is the logic of epidemics in which every possible world is infected by a virus whose transmission is symmetric and transitive.

As for Godel's argument, if we take the special case of his argument in which the positive properties P are taken to be the properties that are true for every possible individual, i.e by taking

$P(\psi) := \Box \forall x, \psi (x)$

and if we replace axiom A1 above with

$P(\psi) \equiv \neg N(\psi)$

where

$N(\psi) := \Diamond \exists x, \neg \psi (x)$

Then i expect that the resulting argument reduces to a trivial tautology of S5 in which all individuals are infected by the godliness virus.

I don't think it has anything to do with mathematics. This is perhaps clearer if we don't consider the button to turn the lamp on and off but instead consider it to alternate between two or more colours.

What number would you assign to the colour red, and why that? What number would you assign to the colour blue, and why that? Shall we use e and i, because why not?

The logic of the lamp just has nothing to do with numbers at all. — Michael

Yes, and that sounds identical to the philosophy of intuitionism :) It doesn't matter what type of object is associated with the lamp's output.

By contrast, Benecerraf et al argue along more classical lines, by defining an abstract completion of the sequence that doesn't contradict Thompson's premises, but which requires changing the original problem by adding an point at infinity to accommodate such a completion.

Thompson's views sound intuitionistic, in that he didn't apparently consider the "completion" of button-pressing to have a meaningful answer in relation to his thought-experiment. Indeed, if one formally treats Thompson's Lamp as being an unfinishable choice-sequence, which is an object equivalent to what computer-science calls a "stream", then Brouwer's weak axiom of continuity explicitly forbids the construction of any theorem that postulates a property of Thompson's stream that isn't decidable in a finite number of steps. Indeed, Brouwer can be understood as inventing intuitionism to explicitly forbid the informal interpretation of total functions as representing super-tasks, which are concepts that are incompatible with temporal intuition.

By contrast, Benacerraf changed the goal posts by giving Thompson's Lamp a formal treatment from the perspective of classical analysis, in which he interpreted Thompson's Lamp as being an incomplete description of an extended abstract function with a point at infinity, whose value can be chosen as being continuous with some property of the original sequence. However, Bencerraf's formal interpretation doesn't have a temporal interpretation in the sense demanded by the intuitionists. And his methodology runs into problems in situations where the function to be completed has conflicting notions of continuity, as in The Littlewood-Ross Paradox.

But recall that Wittgenstein regarded the ordinary meaning of "to know" to not imply infallibility, in the sense that even if a fact P necessarily implies another fact Q, "knowing that P" does not necessarily imply Q.

This stems from his epistemic consideration that in a literal sense nothing is knowable in the sense demanded by a philosopher. And yet he appreciated that everyone including himself ordinarily use the verb "to know" all the time. Therefore he concluded that the ordinary meaning of "to know" isn't an insinuation of ideal knowledge.

If Moore's knowledge of his hands is interpreted in that light, then had Moore later discovered that he didn't actually have hands, his discovery wouldn't contradict his earlier ordinary claim to "know that he had hands"

I imagine that a supertaskist might agree with the formalist or intuitionist that supertasks aren't mathematically or logically defensible, whilst nevertherless insisting that reality contains supertasks, by arguing that mathematical logic is the map rather than the territory, and by conceding that they are only using mathematics as a descriptive tool for expressing their beliefs, rather than as a prescription for justifying their a priori beliefs in super-tasks.

For instance, a supertaskist might appeal to the fact that one cannot say how many moments of time has passed during a minute, or how many physical operations took place in one's computer to sum 1 + 1, and they might appeal to this inability to measure, divide and count experience or events as grounds for being open minded to the idea that space-time is a literal continuum.

But in that case, how does the supertaskist propose identifying what isn't a super-task? If super-tasks are to have empirical meaning and inferential value, the supertaskist must delineate task from super-task, but how can they delineate them on a non-ad hoc basis?

The inability for dimensionless points to be reconciled with the continuum is what motivated Whitehead's point-free geometry, a precursor to the field of Pointless Topology, as for instance formalised using Locales whose distributive law characterizes the meaning of a "spot". (It might be useful to test this law in relation to the SB tree, for both the truncated and infinite version).

See that phrase, "perfect information"? That's why I say formalism attempts to do the impossible. In other words, it assumes an ideal which cannot be obtained, therefore it's assumption is necessarily false. — Metaphysician Undercover

Perfect information isn't an assumption of formal reasoning, rather it is regarded to be a necessary condition of the meaning of "formal" reasoning in that it is by definition finitely deducible and does not require appealing to unformalized intuitions about infinite and ideal objects. Most importantly, the condition of perfect information ensures that formal reasoning cannot interpret an expression such as {1,2,3,...} as representing an abbreviation of some ideal object; the former expression must either be formally treated as a finite object of some type, else the expression must be considered illegal.

It is actually by sticking to formal reasoning that the illusion of the ideal is never obtained. The opposite impression is due to Platonists disguising themselves as formalists, which might be said to even include Hilbert himself.

Formalism makes the reasonable demand that whatever informal intuitions originally motivated the construction of an axiomatic system, and whatever informal interpretations one might subsequently give to the signs of that system, the methodology of theorem-proving should be purely algorithmic and make no appeal to such intuitions, whether such intuitions be rooted in platonism or in Kantian intuition.

I view formalism as a form of Platonism. It's a Platonist game in which the participants deny their true character, that of being Platonist. Notice "perfect information" is the foundational feature of Platonist idealism. That perfection is the only thing which supports the eternality of Platonic ideals. So formalism and Platonism are really just the same thing, even though the formalists will claim otherwise. — Metaphysician Undercover

The irony of Hilbert, is that his formalism ultimately led to the rebuttal of his own informal intuitions about infinity, namely his presumption that a closed axiomatic system must possess a finite representation of it's own consistency. Had Hilbert better understood the implications his formalism, and especially the finite formal meaning of The Law of Excluded Middle which he apparently accepted for instrumental purposes, then Godels incompleteness theorem might not have come as a shock to him. It is evident that Hilbert was a methodological formalist who didn't mean to insinuate that mathematics was a meaningless game void of semantics, but only that the terms used to denote sets, formula and constants shouldn't require interpretation for the purposes of theorem proving. Unfortunately, his intuitions misled him regards to the outcome of his formal program.

If we inspect the finite activity of theorem proving in a formal system, we see that every term that is informally interpreted as denoting an "infinite object" only possesses finite conditions under which the term is introduced into a theorem and under which the term is eliminated from a theorem.

Different formal systems can be regarded as differing only in regards to their ability to distinguish types of finite object. E.g Intuitionism that formalizes choice-sequences can distinguish uncompleted finite sets from ordinary finite sets, whereas ZFC as a theory of first-order logic can only distinguish finitely defined functions from finite sets - so whilst ZFC might be informally said to be a theory about "infinite sets", this isn't the proof-theoretic formal meaning of ZFC, and so a formalist is free to reject the platonic myths that surround ZFC.

You can say that I have a problem with formalism, because I do. Like claiming that accepting certain axioms qualifies as having counted infinite numbers, formalism claims to do the impossible. — Metaphysician Undercover

Formalism as a philosophy considers mathematics to be reducible to a finite single-player sign game of perfect information in which proofs refer to deterministic winning strategies, and hence Formalism does not support the Platonic interpretation of abstract mathematics as denoting actually infinite objects, whatever the formal system concerned.

So I think your problem is actually with Platonic myths that have become psychologically wedded to innocent formal definitions, and in particular the formal definitions of limits and total functions that are ubiquitously misinterpreted in both popular and scientific culture as denoting a non-finite amount of information, E.g as when the physicist Lawrence Krauss misleads the public with nonsense about the physical implications of Hilbert Hotels.

There is a fundamental problem with identifying supertasks with series limits — sime


This is the kind of mistake that Benacerraf makes in his response to Thomson, as explained here.

The lamp is not defined as being on or off at particular times; it is turned on or off at particular times by pushing a button.

This is an important difference and is why so many "solutions" to Thomson's lamp (and other supertasks) miss the point entirely.

If the lamp is turned on after 30 seconds then, unless turned off again, it will remain on for all time. This is why if you claim that supertasks are possible then you must be able to give a consistent answer as to whether or not the lamp is on or off after 60 seconds. If you cannot, because no consistent answer is possible, then this is proof that the supertask is metaphysically impossible.

It is necessary that the lamp is either on or off after 60 seconds, and for it to be either on or off after 60 seconds it is necessary that the button can only been pressed a finite number of times before then. — Michael

My impression of Benacerraf is that he is defining Thomson's Lamp as a boolean valued function

$l : \overline {\mathbb {N}} \rightarrow \mathbb {B}$

on the domain of the extended natural numbers $\overline {\mathbb {N}}$ which introduces an additional point $\omega$ at "infinity", and then arguing that the value at $\omega$ can be chosen arbitrarily and independently of the function's limiting value, if any. But if this the case, then he isn't engaging with Thomson's argument and has merely shifted the goal posts to declare victory in an incomparable axiomatisation.

But the point about Frege's Law Vb also applies to the extended natural numbers; Thompson's lamp when defined as the function $l$ has a domain consisting of two definite and maximally separated points 0 and $\omega$ and a number of points between 0 and $\omega$ that is intensionally described as being countably infinite. However, if Frege's Law Vb is rejected for reasons mentioned previously, then although $l$ still has the aforementioned intensional properties, it does not possess an extensionally well-defined number of points, in which case it cannot be considered to represent the metaphysical notion of a supertask.

Essentially, mathematical analysis will fail to persuade unless one is already a true believer of supertasks.

There is a fundamental problem with identifying supertasks with series limits, namely the fact that literally infinite summations are not expressible in calculus, given that they cannot be written down.

A formalist is free to use the name "1/2 + 1/4 + ..." to denote 1, but the formalist cannot interpret "1/2 + 1/4 + ..." as an expression implicitly representing part of an infinite summation, because the formalist considers expressions to have no meanings other than being finite states of a syntactical parser when proving a theory in a finite number of steps.


Frege fell into a similar trap as the supertaskers in the Grundgesetze when he proposed his law V. He wanted there to be a one-to-one correspondence between every function and it's representation as a table of values, even in the case of functions with infinite domans. So he proposed Basic Law Vb with disastrous consequences:

{x∣Φx} = {x∣Ψx} → ∀x(Φx ↔ Ψx).

To a finitist or potentialist, Law Vb can be interpreted as introducing fallacies of induction into Set Theory, since they will likely interpret the sets-as-extensions on the left hand side as denoting a finite amount of observable information, and they will likely interpret the function on the right-hand side as denoting an unbounded amount of implicit information, meaning that they cannot regard Law Vb to be a reliable rule of induction. Furthermore, according to their reasoning Law Vb cannot be regarded as constituting a definition of the right hand side, unless one gives up the idea of functions having infinite domains).

Wittgenstein himself warns in the preface that PI isn't a very good book and not the book he intended to write. The unfortunate consequence of it not being a good book, and yet being a book of tremendous importance for analytic philosophy, is the necessity of gatekeeping and elitist assholes, partly in order to rectify commonplace misunderstandings of Wittgenstein that were promulgated in the secondary literature by a significant proportion of the previous generation of gatekeeping and elitist assholes.

You first need to distinguish evidence of Christianity from interpretations of "Christianity", in order to clarify the extent to which your argument is grammatical and theological rather than inferential.

Do you really wish to argue that mystical visions are externally related to Christian concepts and present inferential evidence that those Christian concepts denote 'facts'? For how could such an argument ever get off the ground?

didn't think he proposed a solution. Rather, it was an example to show that it is impossible to complete a supertask. — Michael

Yes, in other words rejecting iii), namely the idea that one can finish counting an infinite sequence.

Let's first remember the fact that the limit of a sequence isn't defined to be a value in the sequence.

Re : The Cauchy Limit of a Sequence

"When the values successively attributed to the same variable approach indefinitely a fixed value, eventually differing from it by as little as one could wish, that fixed value is called the limit of all the others"

A converging sequence might eventually settle on value equal to its limit, but even then the two concepts are not the same. So it doesn't matter whether we are talking about Thompson's Lamp, or merely a constant sequence of 1s. In either case, a limit, if it exists, doesn't refer to any position on the sequence, rather it refers to a winning strategy in a type of two-player game that is played upon the "board" of the converging infinite sequence concerned.

So it make no literal sense to consider the value of an unfinishable sequence at a point of infinity, so the meaning of a "point at infinity" with respect to such a sequence can at best be interpreted to mean an arbitrary position on the sequence that isn't within a computable finite distance from the first position. In the newspeak of Non Standard Analysis, such a position can be denoted by a non-standard hyper-natural number, meaning an ordinary natural number, but which due to finite limitations of time and space cannot be located on the standard natural number line.

As for the OP, its triad of premises are inconsistent. For only two of the three following premises can be true of a sequence

i) The length of the sequence is infinite.
ii) The sequence is countable
iii) The sequence is exhaustible

For example, Thompson's proposed solution to his Lamp paradox is to accept (i) and (ii) but to reject (iii). Whereas solutions to Zeno's Paradox tend to start by accepting (iii) but reject the assumption that motion can be analysed in terms of a countably dense linear order of positions, either by denying (i) (namely the assumption that the sequence of positions is infinite, which amounts to a denial of motion) or by denying (ii) (namely the assumption that motion can be used to count positions, for example because the motion and position of an arrow aren't simultaneously compatible attributes).

Sure. That is indeed a different take. I'm taking what I like to think of as a traditional scientific approach, otherwise known as a reductionist materialist approach. Like anyone in this field, I'm driven by a particular set of beliefs that is driven by little more than intuition - my intuition is that reductive scientific methods can explain consciousness - and so a big motivation -- in fact one of the key drivers for me - is that I want to attempt to push the boundaries of what can be explained through that medium. So I explicitly avoid trying to explain phenomenology based on phenomenology. — Malcolm Lett

Consider the fact that traditional science doesn't permit scientific explanations to be represented or communicated in terms of indexicals, because indexicals do not convey public semantic content.

Wittgenstein made the following remark in the Philosophical Investigations

410. "I" is not the name of a person, nor "here" of a place, and
"this" is not a name. But they are connected with names. Names are
explained by means of them. It is also true that it is characteristic of
physics not to use these words.

So if we forbid ourselves from reducing the meaning of a scientific explanation to our private use of indexicals that have no publically shareable semantic content , and if it is also assumed that phenomenological explanations must essentially rely upon the use of indexicals, then there is no logical possibility for a scientific explanation to make contact with phenomenology.

The interesting thing about science education, is that as students we are initially introduced to the meaning of scientific concepts via ostensive demonstrations, e.g when the chemistry teacher teaches oxidation by means of heating a testtube with a Bunsen Burner, saying "this here is oxidation". And yet a public interpretation of theoretical chemistry cannot employ indexicals for the sake of the theory being objective, with the paradoxical consequence that the ostensive demonstrations by which each of us were taught the subject, cannot be part of the public meaning of theoretical chemistry.

So if scientific explanations are to make contact with phenomenology, it would seem that one must interpret the entire enterprise of science in a solipsistic fashion as being semantically reducible to one's personal experiences... In which case, what is the point of a scientific explanation of consciousness in the first place?

Let S denote the set of stairs, let N denote the standard natural numbers and let N* denote the nonstandard numbers. We can model the cardinality of S, which is equivalent to the height of the top of the staircase, by using a non-standard natural number h* from N*. Lets assume

i) There does not exist an injection N --> S
ii) There exists a surjection I ---> S, where I is a subset of N.

Condition i) represents the hypothesis that we do not know how many stairs there are, or equivalently that we cannot know the height of the top stair due to assuming that we will never reach the bottom of the staircase.

Condition ii) represents the physically plausible situation that although we cannot count the stairs, there cannot be more stairs than some finite but unboundedly large subset of the natural numbers.

In other words, we are assuming that S is subcountable.

Let s(n) denote the n'th stair that is visited when descending. Using this order of descent on S, we have a total function S --> N* describing the height of each stair as a non-standard natural number, namely

s (0) => h*
s(1) => h* - 1
s(2) => h* - 2
..

which when written directly in terms of the indices denoting the order-of-descent is a function f

f : N --> N* :=
f (n) = h* - n*


This function describes an infinite descent in N*, and is paradoxical because

1) Every nonstandard natural number e* that is in N* corresponds to some standard number e in N, and vice-versa.

2) We have defined an infinitely descending chain of non-standard natural numbers in N*.

The paradox is resolved due to the fact that the order-of-descent we are using when descending the "infintie staircase" from the top has no recursively definable relationship in terms of the order of ascension when climbing the staircase from the bottom; although Peano's axioms rule out the existence of non-wellfounded subsets for recursively enumerable subsets of the natural numbers, our subset isn't recursively enumerable in terms of those axioms, and is therefore an external subset that cannot be talked about by Peano's axioms.

Sorry Fishfry.

On further reflection the infinite sided die shouldn't need a choice axiom in its construction (e.g a sphere can be painted by working clockwise and outwards from a chosen pole - since there is an algorithm choice isn't needed). But then what of the idea of rolling said die an actually infinite number of times? That surely is equivalent to choice, assuming the rolls are random.

The natural numbers are well ordered in their usual order. — fishfry

Yes, that is true, by Peano's inductive construction of the natural numbers. And a well-order is usually assumed for an infinite sided die, in spite of its construction lacking an inductive specification (for which side should be assigned what number?) - So the assumption of a well-ordered infinite sided die that lacks an inductive definition is the same as a countably infinite set of objects equipped with the axiom of countable choice.

For what it's worth, the fact that we can't put a uniform probability measure on the natural numbers doesn't mean they have to be "all the same number." They're all different numbers. And I can't understand the idea you're getting at. — fishfry

I took the idea to mean that the faces of an infinite die isn't a well-ordered set, unless the Axiom of Countable Choice is assumed. If this axiom isn't assumed, then the sides of the die can only be ordered in terms of their order of appearance in a sequence of die rolls, which implies that unrolled sides are indistinguishable.

I'm quite fond of this potential infinity solution and believe it may be the correct direction to pursue.

However, the die in the paradox possesses an actually infinite number of sides (the set of sides is Dedekind-infinite). What more needs to be said to argue that such a die cannot exist? — keystone

The first problem is one logical inconsistency. In Kolmogorov's treatment, the axioms exclude the proposition; if one introduced such a die as a new axiom, the system wouldn't be consistent. Whereas in my above (very rough) proposal, A Dedekind infinite set is measured directly in terms of its definition rather than in terms of it's cardinality,but which in turn implies that it has lower probability than its subsets, violating additivity, (Here I am assuming that we want to use standard rules for mapping distributions from one set to another. I'm not actually sure if there might be some other workaround than banning Dedekind-infiniteness).

The second problem is one of motive. Is the motive good enough? Consider what it means to say that the Natural Numbers are Dedekind infinite. In type theory, it refers to an object N with an arrow
1 + N --> N that has an inverse ( here 1 denotes zero, and + indicates disjoint union, and the arrow is the successor function). A standard computational reading of this arrow is that it conveys the fact that one can count upwards from zero to an arbitrary finite number of one's choosing and then count downwards to return to zero. In a nonstandard reading one is also allowed to count from an arbitrary position that cannot be reached from zero. But in either case, the arrow doesn't have the extensional significance that set theorists like to assume. That is to say, the arrow doesn't imply that "every member of the natural numbers exists prior to it being counted" , rather the arrow is used to construct as many members as one desires. In summary, we can say that Dedekind-infiniteness is a type of rule that can be used to generate Dedekind-finite extensions of any size that can be freely extended as and when one desires, by applying the rule once more.

In the case of an infinitely sided die, if the die can only be rolled a finite number of times, then its trajectory of outcomes is equivalent to the trajectory of some Dedekind-finite die that by definition is guaranteed to possess an arbitrary but finite number of unrolled sides after the final roll of the die. Is rolling the die a Dedekind-infinite number of times extensionally meaningful? Not according to the functional interpretation of Dedekind-infiniteness, which deems the previous analysis sufficient for the philosophical analysis of the fall of man paradox.