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are they infinite numbers ?

muxol
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Posted 07/24/09 - 01:05 PM:	#21
AKG wrote: What's the difference between a "completed" infinity and an "incomplete" infinity, and can this difference be stated in mathematical terms? The difference is that one is intuitionistically and constructibly acceptable/coherent while the other is not. One can be (mentally, for the intuitionist) constructed in principle and the other cannot. The difference can be stated in mathematical terms, if you take 'constructible (in principle)' to be a mathematical term. There are mathematically rigorous theories of constructions/types. Does rejecting "completed infinites" mean rejecting the axiom of infinity, or is the axiom of infinity not even expressed in the right language to talk about whether or not infinite sets exist. Rejecting completed infinities means rejecting the axiom of infinity under a particular interpretation, e.g. the one that interprets set-theoretic statements according to classical model-theoretic semantics. Of course constructive and intuitionistic set theories may also have the axiom of infinity, but the axiom is interpreted, and hence means, something quite different from what it does for a classicist. I'm not an algebraist or an analyst, but I've studied these things and they seem to use the concept of sets in the same way that foundationalists do, it's just that they don't study sets in general. But they study their specific sets, and they treat them like sets the same way set theorists treat sets, or so it would seem to me. A great number concerned with foundations are constructivists, and probably most that are not so concerned are not constrictivists. It follows, on a popular view of concepts, that they employ different concepts. E.g. one thinks infinite sets are completed and the other does not, even if they both agree on your definition of infinite set. They will also have different notions of choice functions and well-foundedness. (These are touchy topics in constructive set theory since the usual formulations of the axioms of choice and regularity imply, intuitionisitically, excluded middle.) So what seems to you to be the case is likely false. (Another good example contrasts the set-theoretic concept of number with the categorical-theoretic one, a contrast very much reminiscent of that between platonism and structuralism.) I don't think that's a very commonly held view, and I think much of mathematics already uses the framework and semantics of set theory, I can't see why one would want to switch to category theory. Even in the category theory I've studied, a concerted effort is made to reconcile the contradiction between set theory and say a "naive" approach to category, since things like the category of all groups doesn't exist as a set. I'm interested though, could you elaborate on your point?I suppose it's possible, but why would one do that? There are a number of reasons, which I won't recite here, as to why one might prefer category theory to set theory. Take a look at the SEP article and its references for a start. I'm all in favor of classical set theory and platonism, but I think there are a lot of other coherent/consistent (and even convincing!) views about mathematics and mathematical objects, such that one ought not to dismiss them so easily.

muxol
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Posted 07/24/09 - 01:12 PM:	#22
MoeBlee wrote: With those definitions, the set of natural numbers is the same size as the set of rational numbers, since there is a 1-1 function from the set of natural numbers onto the set of rational numbers. One naive view concerning 'bigness' of sets is the following: x is bigger than y iff y is properly included in some isomorphic copy of x. Probably most people have this conception of bigness, which is they find it odd that the rationals are no bigger, cardinality-wise, than the naturals, for the latter is properly included in the former.

MoeBlee
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Posted 07/24/09 - 03:52 PM:	#23
muxol wrote: One naive view concerning 'bigness' of sets is the following: x is bigger than y iff y is properly included in some isomorphic copy of x. If x is not a system (i.e., a tuple with a carrier set and relations and functions on the carrier set) then what does it mean to say 'an isomorphic copy of x'? muxol wrote: they find it odd that the rationals are no bigger, cardinality-wise, than the naturals, for the latter is properly included in the former. Of course.

muxol
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Posted 07/27/09 - 09:48 AM:	#24
MoeBlee wrote: If x is not a system (i.e., a tuple with a carrier set and relations and functions on the carrier set) then what does it mean to say 'an isomorphic copy of x'? What is the puzzle? The case where X is a set of individuals with an empty family of relations (including functions) on X can be viewed as a special case where X is a structure (e.g. an ordering (A,<)), i.e. a set of individuals with a non-empty family of relations on its carrier set. In the special case we're considering, this turns out equivalent to saying that Y is isomorphic to X iff Y is eqiunumerous with X.

MoeBlee
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Posted 07/27/09 - 09:50 AM:	#25
muxol wrote: The case where X is a set of individuals [...] Y is isomorphic to X iff Y is eqiunumerous with X. You can stipulate that if you like; I have no objection. But then the naive view you mentioned reduces merely to the ordinary set theoretic definition anyway.

muxol
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Posted 07/27/09 - 12:26 PM:	#26
MoeBlee wrote: You can stipulate that if you like; I have no objection. But then the naive view you mentioned reduces merely to the ordinary set theoretic definition anyway. No, it doesn't. Being a subset of a set X equinumerous with Y is not equivalent to being equinumerous with Y.

MoeBlee
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Posted 07/27/09 - 04:43 PM:	#27
muxol wrote: Being a subset of a set X equinumerous with Y is not equivalent to being equinumerous with Y. Of course. What I meant is that this is just equivalent to the ordinary set theoretic notion of 'domination', then just a step away from the ordinary set theoretic notion of 'equinumerosity'. I take it that (in the case where there is no relation or function to examine for homomorphism, but rather we have only 1-1 correspondence to consider) your naive person would observe an injection from Y into a subjset of X (or onto some set itself 1-1 with a subset of X) and conclude that X is bigger than Y. But we already have that notion, except that we say it justifies saying only 'bigger or equal' while 'strictly bigger' requires also that there is no injection from Y that is onto X (or onto any set 1-1 with X). The naive notion is untenable (both for isomorphisms and for mere 1-1 correspondences), as I surmise you agree. I'm merely pointing out that that the naive notion is untenable is seen in a particularly stark way when it involves only 1-1 correspondences (that is, untenable, in the sense that it doesn't give a true alternative, or if it does, then it breaks down when we see that a set may inject into a set yet not be strictly smaller since otherwise a set would be strictly smaller than even itself).

muxol
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Posted 07/27/09 - 08:17 PM:	#28
Agreed. I think the reason Joe Average finds the naive view so seductive is that it works flawlessly for finite sets. But this is just pointing out the obvious.

Vulcan23
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Posted 07/30/09 - 03:48 PM:	#29
[quote=eldorado]Hi, this is my first post. from this topic : I think the answer should be "no" , because I have finite set of numbers in my brain at any time. Is that right ? Things can still exist whether or not you can comprehend them.

DavidHilbert
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Posted 08/09/09 - 04:30 PM:	#30
eldorado wrote: Hi, this is my first post. from this topic : http://www.math.utoronto.ca/mathnet/answers/existence.html I understand that numbers are abstract concepts ( exist in our brains only ). So , are they infinite numbers ? I think the answer should be "no" , because I have finite set of numbers in my brain at any time. Is that right ? Thanks. Euclid proved that there are infinitely many prime numbers, that seems to show that numbers are indeed infinite.

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