Uncountability of the Rationals?
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From: Tony Orlow on 4 Jun 2010 12:16 On Jun 3, 3:55 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <1bb4e64e-9dd5-45a2-8c42-9ffa430f8...(a)c7g2000vbc.googlegroups.com>, > Tony Orlow <t...(a)lightlink.com> wrote: > > > In the same vein, most objectors to transfinitology have a hard time > > identifying where exactly they disagree with the construction of the > > theory, but have at least a few examples of conclusions drawn from it > > which are completely incorrect when approached through other avenues. > > This is partly because mathematicians misrepresent the theory, > > claiming that every conclusion they draw "follows logicallly from the > > axioms." This claim is simply not true, as cardinality is not > > mentioned in the axioms, much less anything about omega or the alephs, > > except for a declaration that something homomorphic to the naturals > > exists, which is ultimately not a set, but a sequence. > > The issue is whether cardinality CAN BE defined within an axiom system, > not whether it is explicitly mentioned in some axiom(s) of that system. > > And it can be. And yet, it does not logically follow from the axioms. It is simply consistent in the sense of not contradicting them. Therefore, other measures of sets can also exist, consistent with the axioms, but inconsistent with cardinality. > > How does TO define 'sequence' without reference to something like the > SET of naturals as indices? It is a set wherein every element is either before or after (not immediately) every other element. That's one possible definition, though you undoubtedly have some objection. Peace, Virgil. Tony
From: Tony Orlow on 4 Jun 2010 12:20 On Jun 3, 11:40 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > kunzmilan <kunzmi...(a)atlas.cz> writes: > > The uncountability of the reals is simply based on the fact, that > > there are more rational numbers than there are the natural numbers. > > This is, of course, utterly butt-wrong. > > There are not more rational numbers than naturals -- that is, |N|=|Q|. > > Even Tony knows that. > > -- > Jesse F. Hughes > > One is not superior merely because one sees the world as odious. > -- Chateaubriand (1768-1848) Hi Jesse - Just because I concede that both sets are countably infinite and therefore of the same cardinality, nevertheless the sparse proper subset of the rationals called the naturals should not be equated in size with its dense proper superset. Tony
From: Tony Orlow on 4 Jun 2010 12:26 On Jun 4, 2:05 am, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jun 3, 3:51 pm, "porky_pig...(a)my-deja.com" <porky_pig...(a)my- > > deja.com> wrote: > > On Jun 3, 1:10 pm, kunzmilan <kunzmi...(a)atlas.cz> wrote: > > > Since in both triangles are more different elements than in the first > > > row of the matrix R, there are more rational numbers than in the set > > > of the natural numbers and the rational numbers are uncoutable. > > So we had Herc claiming that reals are countable. now we have > > kunzmilan claiming the rationals are uncountable. The plot thickens. > > At one point, I'd wanted to introduce phrases such as > "substandard theorists" and "superstandard theorists." The > former would refer to posters like Herc/Cooper who believe > in _fewer_ types of infinity than the standard (sub- means > below -- Herc believes that the infinity of the reals and > the infinity of the naturals are the same.) The latter > would refer to posters such as TO and kunzmilan who > believe in _more_ typpes of infinity than the standard > (super- means above -- these two posters believe that the > infinity of rationals and the infinity of the reals are > in fact distinct). > > This would have emphasized that there are posters who > oppose ZFC from two different sides -- those who believe > that ZFC has too many set sizes and those who believe that > ZFC doesn't have enough set sizes. > > But I've said that I would avoid calling different types > of posters as so-and-so "theorists," and so I will not be > using those labels at all. I only point out what I once > had in mind. Hi Transfer - As always your comments are appreciated. Just so you know, there already exist labels describing the two types of objectors to transfinitology: Anti-Cantorians and Post-Cantorians. I think "substandard" probably gives the wrong connotation, though it doesn't offend me, since I'm a "superstandard" Post-Cantorian. Heh :) Have a nice day. Tony
From: Pubkeybreaker on 4 Jun 2010 13:12 On Jun 4, 12:20 pm, Tony Orlow <t...(a)lightlink.com> wrote: > On Jun 3, 11:40 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > > > kunzmilan <kunzmi...(a)atlas.cz> writes: > > > The uncountability of the reals is simply based on the fact, that > > > there are more rational numbers than there are the natural numbers. > > > This is, of course, utterly butt-wrong. > > > There are not more rational numbers than naturals -- that is, |N|=|Q|.. > > > Even Tony knows that. > > > -- > > Jesse F. Hughes > > > One is not superior merely because one sees the world as odious. > > -- Chateaubriand (1768-1848) > > Hi Jesse - > > Just because I concede that both sets are countably infinite and > therefore of the same cardinality, nevertheless the sparse proper > subset of the rationals called the naturals should not be equated in > size with its dense proper superset. > You have been given an explicit injection from Q to N. If you think it is wrong, explain why. Otherwise, explain this idiocy in which you believe that there are more rationals than integers.
From: Aatu Koskensilta on 4 Jun 2010 13:22
Pubkeybreaker <pubkeybreaker(a)aol.com> writes: > Otherwise, explain this idiocy in which you believe that there are > more rationals than integers. It makes perfect sense to say there are more rationals than integers, for example to mean that there are many reals that are rational but not integral. Similarly, in logic we might say -- and indeed do say -- that PA + Con(PA) proves more theorems than PA, in the sense that the set of theorems of PA + Con(PA) is a proper superset of the set of theorems of PA. What is difficult is to come up with any notion of size that applies to sets in general according to which there are more rationals than integers. This is an interesting problem, and we find in the literature many not at all nonsensical attempts at an answer that respects the various intuitions we have, or some people have, or have had, about size. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |