So you are saying that aleph_0 is not a member of the set of natural numbers yet the number of its members is aleph_0. I am however puzzled how all the members of the natural number set are finite yet it has aleph_0 members. — MoK
I defined the domain D that has this specific property, the number of its members, n, is a member as well. — MoK
Let me ask you this question: Are all members of the natural number set finite? — MoK
how the number of its members could be aleph_0 — MoK
I am however puzzled how all the members of the natural number set are finite yet it has aleph_0 members. — MoK
I am however puzzled how all the members of the natural number set are finite yet it has aleph_0 members. — MoK
D) By continuum I mean a set of distinct points without an abrupt change or gap between points.
A) Assume that continuum exists (assume that D is true)
P1) There is however either a gap between all pairs of points of the continuum or there is no gap
P2) We are dealing with the same point of the continuum if there is no gap between a pair of points
C1) Therefore there is a gap between all pairs of distinct points of the continuum (from P1 and P2)
C2) Therefore, the continuum does not exist (from A and C1) — MoK
What do you mean by this? Do you mean that the set of natural numbers is the set of aleph_0? aleph_0 is a number. How could you treat it as a set?The set of natural numbers is aleph_0. — TonesInDeepFreeze
I don't understand this argument. How could aleph_0 be a number and a set at the same time?Every set is one-to-one with itself. So the set of natural numbers is one-to-one with alelph_0, so the cardinality of the set of natural numbers is aleph_0. — TonesInDeepFreeze
So classical continuum exists. I had to read about second-order logic, first-order logic, and zeroth-order logic trying to make sense of what you are trying to say here. Unfortunately, I don't understand what you are trying to say in the bolded part, the last part of the paragraph. Do you mind elaborating?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. — sime
Do you mind elaborating on what problem you specifically have in your mind?But the definition isn't constructive and is extensionally unintelligible for some of the reasons you pointed out in the OP. — sime
Do you by cut mean the exact position of an irrational number for example? What do you mean by the bolded part?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. — sime
Ok, thanks for the reference.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. — sime
My understanding is that there is no upper bound on the number of points that a continuum can be divided into. I however don't understand whether he agrees with the classical notion of continuum or not. If not, what is his point?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, — sime
Do you mind discussing his results further?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. — sime
So, one cannot define infinitesimal in the classical continuum. Is that what you are trying to say?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. — sime
What do you mean by antimony here?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. — sime
So you are trying to say that introducing infinitesimal can resolve the problem of cut which is problematic for classical continuum.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. — sime
What do you mean by temporal and spatial intuition here?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. — sime
I am reading about point-free topology right now so I will comment on this part when I figure out what you mean with point-free topology.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. — sime
I did read your links partially but I couldn't figure out what filter and ideal are. I don't understand how they resolve the problem of the classical continuum too but I buy your words on it.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. — sime
I directly attacked the continuum in the mathematical sense. The discussion is ongoing but it seems that the classical continuum exists but suffers from problems which are discussed here. The post is very technical and I have problems understanding it though.Are you suggesting this proves real numbers are logically impossible, or are you arguing that there is no valid 1:1 mapping from the set of real numbers to the actual world? — Relativist
Do you mean later (instead of former)? If yes, it would be nice of you to elaborate.I ask, because it fails to do the former. — Relativist
I am looking for proof that the set of natural numbers that each its member is finite has aleph_0 members. — MoK
I don't understand this argument. How could aleph_0 be a number and a set at the same time? — MoK
What do you mean by this? Do you mean that the set of natural numbers is the set of aleph_0? aleph_0 is a number. How could you treat it as a set? — MoK
I am aware of that. To avoid confusion, assume that the cardinality of the set of real numbers is X. How could one show that X is the least infinity namely aleph_0? — MoK
A mistake on my part and I am sorry for that. I should have written: "To avoid confusion, assume that the cardinality of the set of natural (I wrote real instead of natural) numbers is X. How could one show that X is the least infinity namely aleph_0?" — MoK
I didn't ask for a proof that shows that aleph_0 is the least infinity but to show that X is the least infinity namely aleph_0. — MoK
the least infinity namely aleph_0. — MoK
I am looking for proof that the set of natural numbers that each its member is finite has aleph_0 members. — MoK
Do you mean that the set of natural numbers is the set of aleph_0? — MoK
aleph_0 is a number. — MoK
How could you treat it as a set? — MoK
one cannot define infinitesimal in the classical continuum — MoK
Therefore there is a gap between all pairs of distinct points of the continuum" — Relativist
Supertasks have been used to show there isn't such a mapping for some cases. — Relativist
What arguments are you referring to that there is no injection from the set of real numbers into "the world"? — TonesInDeepFreeze
Supertasks have been used to show there isn't such a mapping for some cases. — Relativist
If the real numbers are instantiated in the real world — fishfry
If the real numbers are instantiated in the real world, then questions such as the axiom of choice and the Continuum hypothesis become subject to physical experiment. — fishfry
no physics postdoc has ever applied for a grant to study such matters — fishfry
such questions are so far beyond experimental investigation as to be meaningless. — fishfry
such questions are so far beyond experimental investigation as to be meaningless. — fishfry
Banach-Tarski — fishfry
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