I disagree with your assertion that we must be able to determine which group someone belongs to for there to be two different groups. — Michael

So in your scenario, it is not possible to assign Fred to one of the populations, but you maintain that the distinction is meaningful. That strikes me as absurd.

The same applies to your picture. How could you ever determine that what the chap on the left sees is different to what the chap on the right sees?

This, it seems, might be the core difference between our accounts. You insist that there are private phenomena while apparently agreeing that they make no difference, while I say that since there is no difference, there is no private phenomena.

The symbol we're talking about is this: — Magnus Anderson

That's a group of symbols... so you mean the $f$? And your claim is that the definition
$f: \mathbb{N} \to \mathbb{N}_0, \quad f(n) = n - 1$
"specifies what that symbol can be used to represent", but not that what it represents is not somehow contradictory? OK. Then over to you. If you think there is a contradiction here, it's up to you to show it. Exhibited it as derivations of ⊥.

Hence we come back to what it is for a mathematical object to exist, and the point you seem not to have accepted, that for "classical" maths ∃x P(x) is true iff P(x) is derivable in a consistent formal system - as here. You appear to be rejecting that rejection, while claiming not to reject it. All very difficult to follow. Hence, the impression of an intellectual drift on your part rather than any actual argument.

The first fallacious proof they use to show that N and N0 are of the same size is the observation that, if you add 1 to infinity, you still get infinity. — Magnus Anderson

Where do you think this claim appears in the proof?

The claim “infinity + 1 = infinity” does not appear anywhere in the proofs I have used.

The second fallacious proof they use is grounded in the premise that, if you can come up with a symbol that is defined as bijection between N and N0, it follows that a bijection between N and N0 exists ( i.e. it's not a contradiction in terms. ) — Magnus Anderson

The proof doesn't just "define a symbol for a bijection"; it provides an explicit function:

$f: \mathbb{N} \to \mathbb{N}_0, \quad f(n) = n - 1$

• f is the name of a function.
• ℕ is the set of natural numbers. Here, on convention, ℕ = {1, 2, 3, …}.
• ℕ₀ is the set of natural numbers including 0, i.e., ℕ₀ = {0,1,2,3,…}.
• → is read as “maps to” or “a function from … to …”.

So we can read the definition as:

f is a function from the natural numbers ℕ to the natural numbers including zero ℕ₀ such that for each natural number n, f(n) is equal to n minus 1.

What is defined here is a function, not a symbol. This is a concrete mapping, not a mere linguistic construct, and it suffices to show that a bijection exists.

That's a lie you've been shamelessly pushing forward. — Magnus Anderson

Well, it's not just me...

The definition you suggest cannot be used effectively with infinite sets. But enumeration, that is a surjection, will do everything that can be done with your definition and then do more - quite a bit more - with other sets. So your definition is effectively included in yet surpassed by enumeration.

For anyone keen on a heavy read, The Size of Sets is an Open Logic chapter that goes through most of this. It's a work in progress, so a bit patchy. It goes in to great length concerning enumeration, which is pivotal here.

1) To say that S is larger than S' means that S' is a proper subset of S.
( A definition that applies to all sets, regardless of their size. ) — Magnus Anderson

This is false, since that definition applies only to finite sets. For infinite sets, we need something more. Consider that the even numbers form a proper subset of the integers, and yet we could count the even numbers... a bijection.

The objection that we could not actually count the even numbers because there are two many of them is trivial; we have a function f(n)=2n, that when applied to $\mathbb{N}$ gives us every even number. And we have the inverse, g(n)=n/2, which wen applied to the even numbers gives every integer. If your finitism is such that you cannot see that, I can't help you.

Consider that there are two subspecies of humanity such that what one sees when standing upright is what the other sees when standing upside down. Both groups use the word "up" to describe the direction of the sky and "down" to describe the direction of the floor. Firstly, is this logically plausible? Secondly, is this physically plausible? Thirdly, does it make sense to argue that one subspecies is seeing the "correct" orientation and the other the "incorrect" orientation? Fourthly, if there is a "correct" orientation then how would we determine this without begging the question? — Michael

& , please excuse my interjecting. How would we be able to distinguish between these two populations?

Suppose Fred presents himself to your laboratory, and you are tasked with deciding which population he belongs to. How do you proceed?

I don't see that you can.

And the mistake here seems to be that of presuming there is a private notion of up and down; that is, there is no fact of the matter for Fred to belong to one population rather than the other.

So I'll opt for saying that Michael's scenario is incoherent.

Added: I think, although I haven't worked through it yet, that by treating "up" and "down" as indexicals we could show there to be only one population. Indexicals don’t tolerate private degrees of freedom. To master “up” is to participate competently in a network of practices: standing, pointing, correcting, navigating, explaining, and so on. If two groups are indistinguishable across those practices, then — by the criteria that individuate the concept — they are the same group.

If we would claim there to be two populations, then we must have a way to differentiate them. The set up of the scenario rules out behavioural and functional differences. Pointing out that "up" and "down" are indexical rules out private differences - what's up for me is just what is up for me.

The pull toward “two populations” comes from smuggling in a Cartesian picture: an inner orientation space that could be inverted independently of outer practice. Once that picture is rejected — as both Wittgenstein and Davidson would insist — the multiplicity evaporates.

Yeah... good point. I overstepped.

So in both classical and constructionist maths, for any number we can construct its successor. Ok.

So constructivism will not help Magnus here. He must resort to finitism - the view that why for any number we can construct its successor, we can't thereby construct the infinite sequence $\mathbb{N}$.

Explicitly specifying a function is acceptable as a constructive proof. Constructivism shares some concerns with finitism, but it is not as bonkers stringent. — SophistiCat

I suppose so. I don't see that a constructivist would have issues with f(n)=n-1 or f(n)=n+1. Again, these are not examples with which a constructivist might take issue. They would more typically take issue with LEM, and reductio arguments, and treat infinite collections as potential rather than actual, or as given by generation rules. I've some sympathy for it, after Wittgenstein.

So I think Magnus must be basing his ideas on a finitist intuition. We'll have to see what he says.

Cheers, . It appears we now agree as to almost everything. The flower has many properties, perception makes some of these - colour, smell, shape - salient. Other properties are accessed via background knowledge (life cycle, chemistry, ecology). No single mode of access exhausts an object. We now have no epistemic veil and no private content; public objects anchor meaning; interpretation is world-directed. We both acknowledge the distinction beteween causal and epistemic mediation.

Perception is interpretive, mediated, and embedded in the world — and none of that entails indirectness.

Yes! Magnus's objections are framed as an internal problem with a proof, when they should be framed as external problems with the process being used.

If Magus would be a constructivist or intuitionist here, then he might do well to do so explicitly. That would be a legitimate position. But what we have looks like intellectual drift rather than anything solid.



There's a few ways that a constructivist might proceed. They might reject the usual account of what it is to be a mathematical entity, ∃x P(x) is true iff P(x) is derivable in a consistent formal system. They might instead insist on a constructive approach: To assert “∃x P(x)” one must provide a construction (algorithm, procedure, or finite method) that yields such an x.

So an argument might proceed by rejecting $f: \mathbb{N} \to \mathbb{N}_0, \quad f(n) = n - 1$ as a suitable account of a function, on the grounds that it relies on unrestricted quantification over a completed infinite totality, saying something like "We don't yet have a finite or algorithmic construction of the entire inverse mapping, so surjectivity is not constructively justified.”

The trouble is that for $f: \mathbb{N} \to \mathbb{N}_0, \quad f(n) = n - 1$ we do have the inverse mapping: $g: \mathbb{N} \to \mathbb{N}_0, \quad g(n) = n + 1$. So this won't work here.

This wouldn't be a function on which a constructivist might try to stand their ground. There are, of course, other bits of maths were constructivists interestingly differ from classical maths, and there are some interesting philosophical issues here. But one needs a grounding in mathematics in order to be able to express the difficulties with clarity.



A more eccentric approach, and this is perhaps were Magnus is coming from, might reject infinities altogether. This is the most charitable interpretation Ive been able to work out. If Magnus rejects the very idea of infinite totalities, if he rejects $\mathbb{N}$, then his argument might be made consistent, but at a great cost.

On this view, computing n−1 for any given numeral would be fine, and also computing n−1 for any finite set of numerals, say {2,4,6,8}. But somewhat arbitrarily, a finitist would reject applying n−1 to any infinite set. They in effect accept $\quad f(n) = n - 1$ for any finite n, but not for $\mathbb{N}$.

Importantly, on this view the argument given above would not be invalid, or lead to contradiction, but ill-formed, because it relies on $\mathbb{N}$. (added: It's not even true or false, since these notions do not apply to ill-formed formations )

The cost here is the rejection of succession (roughly, that for every number there is another number that is one more than it; or more accurately, that we can talk about such an infinite sequence); and consequently the rejection of the whole of Peano mathematics*. No small thing.

To be sure, this is how Magnus might have argued, but hasn't.

I, like most folk, enjoy talking about infinity, and so would reject such finitism.

* On consideration, this last isn't quite right. we might accept Peano's definition of succession and still not accept that we thereby construct an infinite set. thanks, .

When I say the function is bijective by definition, I do not mean that bijectivity is explicitly stated, but that it is an unavoidable consequence of the definition. The "proof" consists solely in unpacking what is already implied by the definition, not in adding any further stipulation. — Magnus Anderson

Yep. that's what a proof does.

Not N0 but f(n) = n - 1. That function is a bijection by definition. — Magnus Anderson

Here's the definition again: $f: \mathbb{N} \to \mathbb{N}_0, \quad f(n) = n - 1$

$\quad f(n) = n - 1$ as it stands is not a definition of a bijection. It can't be, because it lacks a domain and a codomain, as provided by $f: \mathbb{N} \to \mathbb{N}_0$

$\quad f(n) = n - 1$ could be applied to any domain, with differing results. With $\mathbb{Z}$ as the domain and codomain also $\mathbb{Z}$, it would be a bijection. If the domain were $\mathbb{N}$ and the codomain $\mathbb{Z}$, bijectivity would again depend on proof, not stipulation.

Yes. It is not explicitly stated in the definition. However, the definition implies it. — Magnus Anderson

An odd thing to say, since making that implication explicit is exactly what the proof presented above does. you treat $\quad f(n) = n - 1$ as if it secretly meant "let $f$ be a bijection defined by $\quad f(n) = n - 1$"; but that is not what is being done. What was done, step by step, was:

1. Define a function by a rule.
2. Specify domain and codomain.
3. Prove that, given those, the function is injective and surjective.

$\quad f(n) = n - 1$ might be bijective, non-surjective, or non-injective depending on the domain and codomain.

It is defined as a bijection. — Magnus Anderson

$\mathbb{N}_0$?

Well, no. It is defined as f(n)=n−1 and then shown to be a bijection. That definition does not mention bijectivity at all. At this stage, the function could turn out to be injective, surjective, neither, or both. Nothing is being smuggled in.

While a square-circle is defined using incompatible properties, there is no contradiction in $\mathbb{N}_0$.

Yep - not small differences. I hope. to get back to our other conversation soon. First cool day in a week so gardening to catch up with.

To some extent your response here also seems pragmatic. — Tom Storm

Well... not quite, although there are simialriteis.

What's absent, amongst other things, is the usual, somewhat naive view that truth is about practicality, that the utility of a sentence is what renders it true, or that there are no true sentences, only more useful ones.

I certainly would not call myself a pragmatist.

The better answer to the question of "what is it to see a ship?" is "I have no idea, but I do. " — Hanover

I think it'd be more informative to answer "Look over there... see that? it's a ship". Show, don't tell. (Edit: Notice that this is public and communal, it presumes that others are involved, as opposed to the solipsism seen in phenomenalism?)

And that's not quiteism. You and I understand what it is to see a ship, because that's what we do. Meaning as use.

Because they indubitably exist — Hanover

Well... we see things, and talk about them and so on - we interact with them and with each other. What place there is for private mental phenomenon in all this is at the very least questionable. You've seen my arguments rejecting qualia for similar reasons.

That word, exist... so often leads to reification.

Wasn't the Wittgensteinian objective to isolate out metaphysical confusion from philosophical inquiry? — Hanover

I'm not privy to Wittgenstein's intentions. I read him as primarily showing that what are thought of as philosophical problems are often, and perhaps always, confusions that can be sorted by rearranging the way we understood them.

(1) your description of direct realism is definitional, not metaphysical. — Hanover

Well, metaphysics is just conceptual plumbing, after all. So metaphysics is "definitional". Btu yes, I'm really not advocating direct realism so much as rejecting indirect realism, together with its reliance on private phenomenon.

(2) ...But to just say the perception then is just part of the process is empty — Hanover

So you would rather a wrong answer here to no answer?

The phenomenonal state remains a mystery, beyond philosophical description. — Hanover

The supposed "phenomenal state" is a large part of the problem. Why take such positing private phenomena as a metaphysical given?

The difference I see in our positions is perhaps in my insistence that the boundaries of philosophical inquiry do not imply anything about ontology. — Hanover

So far as philosophy consists in conceptual clarification, it doesn't presume an ontology. However there are things that we do talk about, so there are ontological ramifications here.

We define a function:

$f: \mathbb{N} \to \mathbb{N}_0, \quad f(n) = n - 1$

• Well-defined: For every $n \in \mathbb{N}$, we have $n \ge 1$, so $n-1 \ge 0$. Hence $f(n) \in \mathbb{N}_0$, and the function is well-defined.
  
• Injective: Suppose $f(n_1) = f(n_2)$. Then
  $n_1 - 1 = n_2 - 1 \implies n_1 = n_2$.
  Hence $f$ is injective.
  
• Surjective: Let $m \in \mathbb{N}_0$. Define $n = m + 1 \in \mathbb{N}$. Then
  $f(n) = f(m+1) = (m+1)-1 = m$.
  Hence $f$ is surjective.

Conclusion: The function $f(n) = n-1$ is a bijection between $\mathbb{N}$ and $\mathbb{N}_0$.

We experience the world through something it is not, phenomenal representation, just as you can experience your appearance through something you are not, a photograph. — hypericin

While addressed to hypericin, this post is for all.

This might be a side-issue, or perhaps the following point is worth making.

There's a line of argument, a form of scientism, that runs something along these lines: the chairs, tables and cups that make up our world are not as we see them but consisting of atoms or quarks or quantum fluctuations or some such; therefore the chairs tables and cups are not the things that make up our world.

Now I hope it's very clear that this line of argument is not only invalid, it is mistaken. That a chair consists of atoms or quarks or quantum fluctuations simply does not mean that it is not a chair.



At least a part of the problem here is that direct realism, as criticised by the indirect realists, is wholly accepted by hardly anybody. The reasons for this are partly historical. When it was noticed that we construct our understanding of the world around us using our brains, folks supposed that this meant we didn't see our world directly. They therefore inferred the existence of philosophers who thought we did see the world directly, and called them "direct realists".

There's also much vagueness concerning what it is to see something indirectly. You didn't see it directly, you saw it through a telescope, or a mirror, or only its shadow. It appears that how we are to understand "direct" perception depends entirely on what it is contrasted with; so of course it is difficult to imagine what "direct perception" is, per se. It's a nonsense, an invention of the defenders of the sort of argument from Ayer that was critiqued by Austin. You can find examples in every thread on perception.



What I would reject here is the idea, incipient in the physiological description of perception, that we do not see the flower, but an image of the flower. The argument being rejected is along the lines of the one given above that there are not really any chairs and tables and cups. It runs along the lines that what we see is not, and here the language gets a bit weird, the "flower-as-it-really-is" or the "flower-in-itself"; what we see is instead a construct built by light and atoms and neural nets.

Now I think this account is wrong, and on two counts. The first is count is the supposition that there is a useful way in which there is a "flower-as-it-really-is" or the "flower-in-itself". This idea relies on it making sense to talk of a flower seperate from our interpretation and construction of the world around us, a flower apart from our comprehension of the world. But our understanding is always, and already, an interpretation, so the "flower-as-it-really-is" or the "flower-in-itself" is already a nonsense.

The second count is the misdirection in thinking that we see the result of the causal chain, and not the flower. We do not see the result of the causal chain, as if we were homunculi; rather, that causal chain just is our seeing the things in our world.

We do not "experience the world through something it is not, phenomenal representation".

Firstly, the word "phenomenal" is doing damage here, by reifying the process of perception, mistaking the process for the result. It presumes, rather than argues, that what we see is the phenomena and not the flower.

And secondly, we do not "experience the world" passively, in the way supposed. We interact with it, we pick up the cup, board the ship, and coordinate all of these activities with others. We do not passively experience the world, we are actively embedded in it.

And none of this is to deny the casual chain that is part of this interaction.

There's a parochial madness here that is pretty sad.

No philosophy. Just a lot of special pleading and tu quoque. — Ciceronianus

The supposed "ideological crisis" is a result of dropping any pretensions of acting ethically, in favour of just openly being inconsiderate, narcissistic twats. Trying to rake back any intellectual dignity from the mess that is the GOP is a lost cause. Intellectual dignity is not on the menu. One cannot have such an "ideological crisis" unless one is committed to at least appearing to have a standing commitment to coherence, justification, or ethical self-understanding. Those pretensions have simply been abandoned.

It’s not that the GOP can’t supply a philosophy, so much as that supplying one would be instrumentally pointless given the current incentives. Attempts to reconstruct "Trumpism" (which is not even a "thing", as the kids say) as a coherent doctrine (national conservatism, post-liberalism, etc.) are absurd; trying to smuggle normative seriousness back into a practice that now explicitly disavows it.

But I would say that.

Carry on.

I very nearly missed your post, yet it's the part of the discussion that I think is novel. Much of the rest has been gone over many times in these fora.

I'll try to be clearer about what it is I think that the Markov Blanket shows. It's to do again with the difference between the causal and the epistemic accounts. A Markov blanket can be placed in different parts of the causal chain with similar results. Consider the causal chain flower - camera - screen - eye - brain. Here are four possibilities:

• Blanket boundary: Around the brain
  
  • Internal states: Brain
  • External states: Everything outside the skull
  • Sensory states: Eye signals
• Blanket boundary: Around the eye
  
  • Internal states: Eye + Brain
  • External states: Screen + Camera + Flower
  • Sensory states: Retinal signals
• Blanket boundary: Around the screen
  
  • Internal states: Screen
  • External states: Camera + Flower
  • Sensory states: Screen pixels
• Blanket boundary: Around the camera
  
  • Internal states: everything behind the camera
  • External states: Flower
  • Sensory states: camera signal

In all three of these, the causal chain remains the same. In the first, the brain "sees" the signal from the eye; in the last, the whole apparatus "sees" the flower; now that's oddly reminiscent of the whole direct/indirect fiasco...

And causally speaking, there's where we can rest. The difference is not in the causal chain, but where one spreads one's Markov blanket.

So, and here we can reject much of the account @Michael has promulgated, since causal mediation does not entail indirect perception.

More anon.

Ok. Nuanced stuff. Noice.

I have to disagree a bit with this:

...dropping the remainder-based role that functions perfectly well in the finite case. — Esse Quam Videri

The "remainder-based role" is not dropped; the use of bijection keeps everything that the alternative has to offer, and adds the ability to deal with infinities. The shift doesn't sacrifice the old inferential roles, it enriches them.

Ok. I'll hold back. We'll see.

Yep, at least the pattern is the same.

Cheers. I'd be interested in your take on my comments regarding formal language. I see it as a refinement of, rather than distinct from, natural language.

Ok. The point of the direct vs indirect realism debate is precisely about the subject’s epistemic relation to objects, not the causal chain that brings that relation about. The causal chain is agreed.

And while they are seeing the image on the screen and they are seeing the ship and they are talking about the ship, each of these has a slightly differing sense, each is involved in a different activity. The first, they might see the screen and talk about how it fits in to the causal chain that leads to them seeing the ship. The second, they see the ship. The last, they fit the ship in to their epistemic background.

The indirect realist sees the causal chain and says that perception is indirect. The direct realist sees the chain and point out that the chain is how we know about the ship. For the direct realist, the chain is the mechanism by which the world shows itself, for the indirect realist, it is a veil hiding it.

Yeah, well, I gather you use mind as a distinct "substance" in your theology, so it works for you there. My rejection of the mind/wold divide is methodological, not just convenient. But this:

This is to mean, if you can jettison the distinction between mental states and external states on the grounds it makes reality easier to comprehend, regardless of whether it comforts with actuality, then you've made it no less logical to insert other preferences into this mix. — Hanover

I again was not able to follow. The fact that mind and world interact I hope we both take as granted, and so ought be suspicious of any doctrine of substances that appears to impede this interaction.

I guess we might also acknowledge two variants on silentism; the one that says there is no further explanation, and the one that in comfortable with lack of congruence.

Australia's health system is far from perfect, but the idea that we would do better to emulate the system in the USA can only be met with derision.

Interestingly, my local government - I live in the Australian Capital Territory - recently took over control of a large religious hospital because of the incompetence of the Catholic administration. Socialism at work, for the benefit of all.

I'm not suggesting that there are not things people with disabilities cannot do. Rather, I'm pointing out that how we talk and think about what folk can't do serves to exacerbate the problem, making other things harder for them to do; and that we can as an alternative refocus on wha they are able to do, to the benefit of all.

I tried to follow that, but failed. See this response from ChatGPT. And Claude, from the same prompt, concluded

This post articulates real philosophical concerns about actual vs. potential infinity, echoing positions from intuitionism and finitism. However, it:

Makes technical errors about what Dedekind-finite infinite sets would be
Misattributes motivations to Hilbert and misrepresents Gödel
Overstates the practical impact on mathematics and science
Presents a minority foundational view as obvious "common sense"

The core intuition—that treating "1, 2, 3, ..." as a completed totality involves a conceptual leap—is worth taking seriously. But the execution here conflates technical and philosophical issues, and the dismissal of modern foundations as "adhoc" ignores their substantial mathematical and philosophical motivation.
Salience: Relevant to foundations and philosophy of mathematics, but overstated regarding impact on working mathematics. — Claud Sonnet 4.5

I'm not sure how to proceed here.

Cheers. Useful stuff. When someone makes such obvious mistakes, it's probably not worth giving detailed responses, because chances are they will not be able to recognise or understand the argument. The result will be interminable.

I think part of what’s driving the disagreement here is that two different notions of “same size as” are in play, and they come apart precisely in the infinite case. — Esse Quam Videri

Yes, but this far too charitable. There are compelling reasons for rejecting Magnus's account. The notion of "same size" he work with is inadequate to deal with infinities coherently - using it results in inconsistencies.

Here's a formalisation of Magnus's account.

• Proper Subset Principle
  If $A \subset B$ and $A \neq B$, then $A$ is smaller than $B$.
• Subtraction Principle
  If $B \setminus A \neq \varnothing$, then $B$ is larger than $A$.
• Transitivity of Size
  If $A$ is smaller than $B$ and $B$ is smaller than $C$, then $A$ is smaller than $C$.

These principles are all valid for finite sets.

Let's look at a few contradictions that result.

Contradiction 1: ℕ vs Even Numbers
Let

$\mathbb{N} = {1,2,3,4,\dots}$
$E = {2,4,6,\dots}$

• $E \subset \mathbb{N}$ and $E \neq \mathbb{N}$
  ⇒ by (N1), $E$ is smaller than $\mathbb{N}$.
  
• $\mathbb{N} \setminus E = {1,3,5,\dots}$ is infinite
  ⇒ by (N2), $\mathbb{N}$ is larger than $E$.

But define the pairing:

$f(n) = 2n$

This is a one-to-one correspondence between $\mathbb{N}$ and $E$.

So:

• $E$ and $\mathbb{N}$ are the same size.
• $E$ is strictly smaller than $\mathbb{N}$.

Thus:

$E \lt \mathbb{N} \quad \text{and} \quad E = \mathbb{N}$

This violates antisymmetry.

Contradiction 2: ℕ vs ℤ
Let

$\mathbb{N} = {1,2,3,\dots}$
$\mathbb{Z} = {\dots,-2,-1,0,1,2,\dots}$

• $\mathbb{N} \subset \mathbb{Z}$, proper subset
  ⇒ by (N1), $\mathbb{N} \lt \mathbb{Z}$.
  
• $\mathbb{Z} \setminus \mathbb{N}$ is infinite
  ⇒ by (N2), $\mathbb{Z} \gt \mathbb{N}$.

But define a pairing:

$0 \leftrightarrow 1,\quad<br /> -1 \leftrightarrow 2,\quad<br /> 1 \leftrightarrow 3,\quad<br /> -2 \leftrightarrow 4,\dots$

So:

• $\mathbb{N}$ and $\mathbb{Z}$ are the same size.
• $\mathbb{N}$ is strictly smaller than $\mathbb{Z}$.

Again:

$\mathbb{N} \lt \mathbb{Z} \quad \text{and} \quad \mathbb{N} = \mathbb{Z}$

Contradiction.

Contradiction 3: Self-Subtraction
Let $A = \mathbb{N}$.

Partition $A$ into two disjoint infinite subsets:

$A = E \cup O$

where

$E = {2,4,6,\dots}$
$O = {1,3,5,\dots}$

By (N1):

• $E \lt A$
• $O \lt A$

But:

$A = E \cup O$

So $A$ is the union of two sets each strictly smaller than $A$.

This is impossible under the naïve size rules, which are now mutually inconsistent.

Contradiction 4: Hilbert’s Hotel
Let hotel $H$ have rooms ${1,2,3,\dots}$, all occupied.

Define:

$f(n) = n+1$

This moves each guest up one room, freeing room 1.

• No guests are removed.
• A new room becomes available.
• The hotel is both “the same size” and “larger”.

Under subtraction-based size:

• Adding capacity without increasing size is impossible.
• Removing nothing yet gaining space is impossible.

The governing rules of “size” break down.

Conclusion
Once infinite sets are admitted, the principles:

• proper subset ⇒ smaller,
• remainder ⇒ larger,
• antisymmetry and transitivity,

cannot all be maintained. The naïve notion of “same size” does not merely yield counter-intuitive results — it generates outright contradictions.

This is the sense in which the mathematical objection applies: the concept fails to define a coherent ordering on infinite collections.

Thanks to ChatGPT for help with the formatting, but even so the time taken to respond to the sort of nonsense promulgated by maths sceptics is far more than the net benefit.

Once that distinction is on the table, the question isn’t really “who is right,” but what we want the concept of “same size” to do in this context. Mathematics answers that one way; ordinary language answers it another. — Esse Quam Videri

The question is, "who is right?", and the answer is, the contradictions above show that Magnus' ideas cannot be made consistent. Formal language is nothing more than tight use of natural language - it is not unnatural. What is shown by the contradictions is not a conflict between natural and formal languages, but a lack of adequate tightness in Magnus's argument. Magnus’s argument lacks sufficient precision to handle the case he wants it to handle.

Notice also that the arguments stand alone, they are not appeals to authority.

The correct diagnosis is not conceptual pluralism, but logical failure.

ChatGPT will summarise a discussion:


Summarise the argument and responses at https://thephilosophyforum.com/discussion/16296/disability/p1 and the next few pages, by topic.


Result: https://chatgpt.com/share/6962116e-63d8-800f-ad10-77d861ed4b8a


This seems to have given a reasonable summary of the conversation and offered to do the next few pages.

Sorry, I hadn't noticed this:

That a bijective function exists, cretin, does not mean that the two sets can be put into a one-to-one correspondence. — Magnus Anderson

:lol:

Oh, well. :roll:

Reading isn't thinking. — Magnus Anderson

Nor is your making shit up.

Reading a maths book isn’t just passive; it’s fuel for precise thinking, especially when you’re debating infinite sets. It shows how folk have thought about these issues in the past, and the solutions they came up with that work.

Your responses are now a bit too sad to bother with. Thanks for the chat.

What you provided is the definition of the countable infinity. That's not the same as infinity. — Magnus Anderson

Well, it's one infinity amongst a few others...

If you want to prove that my definition is false — Magnus Anderson

Your "definition" of infinity is not a definition of infinity. It's not false, it's just an intuitive approximation.

Simply asserting that my definition is a heuristic that is useful for intuition is not an argument. — Magnus Anderson

Yep. So I went the step further, presenting one of the standard definitions.

That goes against what Cantor said. — Magnus Anderson

It seems then that you haven't understood Cantor, either.

And I am pretty sure you won't be able to prove it — Magnus Anderson

A bijection exists between N and A — e.g., $f(n) = \frac{1}{n+1}, \quad n \in \mathbb{N}$


You really should take 's advice and read a maths book.

Let A be a finite set that is { 1, 2, 3, ..., 100 }.
Let B be a finite set that is { 1, 2, 3, ..., 99 }. — Magnus Anderson

Matching one to one from the left, the one left out is the 100. :meh:

With your
A = { 1/2, 1/3, 1/4, ... }
and
N = { 1, 2, 3, .. . }

There isn't last element. Nothing is left out.

They aren't the same size. The set of even numbers has two times smaller. Doesn't matter what Cantor and mathematical establishment say. They aren't reality. — Magnus Anderson

Yep, the evens only has every second number, so it must be half the size... Thanks for the giggle!

...a number that is larger than every integer... — Magnus Anderson

...is not the definition of infinity. “Larger than every integer” is a heuristic, useful for intuition, but the mathematical definitions depend on limits or cardinality. Something like:

S is countably infinite ⟺∃f:N→S that is bijective (one-to-one and onto).

A heuristic for sets is the Infinite means the set never ends; there’s no last element. That allows for sets with transfinite elements.

And adding four to an integer is still an integer. — Magnus Anderson

Sure. Infinities are not integers.

If "add" means "increase in size" — Magnus Anderson

But it doesn't.

Adding four to infinity is still infinity.

By definition, to add an element X to an existing set of elements S means to increase the size of that set. — Magnus Anderson

Not for infinite sets. For obvious reasons.

ℕ and ℕ ∪ {0} really are the same size
Take:

ℕ = {1,2,3,…}
ℕ₀ = {0,1,2,3,…}

here:

f(n) = n - 1

This is:

• injective (no collisions)
• surjective (every element of ℕ₀ is hit)
• Total

That is a proof of equal cardinality. Nothing is “pretended”.
The fact that this offends finite intuition is exactly what “infinite” means in modern mathematics.

You should get on well with Meta.

I'll leave you to it.

The topic attracts cranks.

See The Enumeration of the Positive Rationals

It should be pretty clear.