An uncountable countable set
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From: MoeBlee on 25 Aug 2006 12:44 Tony Orlow wrote: > No, it all rests on the notions of identity and equality. As Leibniz > pointed out, when the properties of two objects are all exactly the > same, then they are the same object. So, when we say two numbers are > equal, that means all properties of the two are equal. Ha! The fallacy of reversing implication right there! An example of just about the most basic fallacy. No, the indiscernibility of identicals does NOT imply the identity of indiscernibles. You need both implications; you can't derive one from the other. And, in first order logic, one direction can be posited only in the semantics not in the axioms. MoeBlee
From: MoeBlee on 25 Aug 2006 12:46 Tony Orlow wrote: > Yes, and the universe is consistent by definition, so math should be > consistently overall as well. Unless the universe is a set of sentences, the notion of consistency does not even apply. > Yes, personally I want concepts of infinity to be compatible with the > rest of math, as a logical extension. There is something amiss in set > theory in this respect. "Logical extension". You have no idea what you're talking about. MoeBlee
From: MoeBlee on 25 Aug 2006 12:50 Tony Orlow wrote: > > > > A set is infinite if it is not finite. A set is Dedekind infinite if > > > > it can be mapped to a proper subset of itself, and it is Dedekind > > > > finite if it is not Dedekind infinite. When we assume the axiom of > > > > choice, the two notions are identical. Without that axiom there > > > > can be infinite sets that are Dedekind finite. Do you want to know > > > > more about set theory? > > > > > > > > Now, using, this terminology (pretty standard), what is a "finite but > > > > unbounded" set? > > > > > > Using that system, the phrase is senseless, but that system is not the > > > real universe. It's a concoction. > > > > Well, that is just a statement that (in my opinion) is senseless. > > Only in the context of the Dedekind definition of infinity, How rude. The man just explained to you the difference between Dedekind infinite and infinite, and you just came back with an ignorant remark that is ignorant for the very fact that it ignores the explanation just given you. MoeBlee which one is > not obligated to consume wholesale. Until one can prove that transfinite > set theory is "correct", no one is obligated to accept the theory at all. > > TOny
From: Tony Orlow on 25 Aug 2006 12:42 David R Tribble wrote: > Tony Orlow wrote: >>> Please >>> show where I am being inconsistent, not with set theory, but within my >>> own assumptions. Remember, I don't claim to believe in transfinite set >>> theory, and don't intend to be consistent with it. > > David R Tribble wrote: >>> The problem is that your own assumptions are not consistent with >>> each other - they don't form a coherent theory. Many of these >>> inconsistencies have been pointed out to you before, but you choose >>> to not believe them or simply to ignore them. > > Tony Orlow wrote: >> It's easy to say that without mentioning any specific inconsistencies. >> >> In fact, mapping the naturals in [1,Big'un] to the reals in [Lil'un,1] >> using the mapping function f(x)=x/Big'un yields Ross' Finlayson numbers, >> and is perfectly consistent with IFR. Not only do we obviously have >> Big'un^2 reals on the line because we have the sum of Big'un unit >> intervals each containing Big'un reals, ... > > That implies that there are BigUn naturals in the real number line, > so that |N| = |R|. You state this as though it's a fact, but what is > your proof? > What? No. I mapped the reals in [Lil'un,1] (Lil'un is successor to 0, in the Finlayson system. He calls it iota, which is finite) to the naturals in [1,Big'un] (the real line). The reals in the unit interval are the image of the naturals on the entire line. You are mistakenly applying the standard cardinalistic fact that the number or reals in [0,1] is the same as the number of all reals. That is false in my system. The set of all reals is the sum of all sets of reals in each unit interval, infinitely (Big'un times) greater. Tony
From: Tony Orlow on 25 Aug 2006 12:51
David R Tribble wrote: > Tony Orlow wrote: >>> But Monsieur, what about the injection from P(N) into N, via the bit >>> strings which denote set membership, each of which also corresponds to a >>> binary natural? Tsk, tsk. Mustn't forget that one! Remember, the only >>> set which doesn't map is the entire set, and that maps to the largest >>> natural, that is, ...1111 with all bits in finite positions. > > David R Tribble wrote: >>> ... as well as all the infinite subsets of N. You keep forgetting >>> about those, don't you? > > Tony Orlow wrote: >> You must be forgetting that, given that all bit positions are finite, >> even your countably infinite bit strings only can represent finite >> values, ... > > That's false. It's also irrelevant, since no natural is represented by > a countably infinite bitstring. > That's an empty claim. If a finite whole number with successor isn;t a natural, what is? > >> ... and since they also all have distinct successors and >> predecessors, your claim that they are not natural numbers is rather >> unjustified, wouldn't you say? > > You appear to be changing the subject. Not at all. You claim countably infinite bit strings do not represent finite naturals, but they are provably finite in value and obvious successors using the normal rules for digital successorship. > > I was talking about your alleged bijection between N and P(N), > which does not exist. It does not exist because the members of > N (which are all finite bitstrings) do not map to _any_ of the > infinite subsets of N, which comprise the bulk of the members > of P(N). > Those are countably infinite strings, which are actually finite but unbounded, yielding finite but unbounded values. There's a big grey area in standard theory which isn;t recognized in this respect. > >> You notice that I called ....1111 the largest natural, don't you? > > Yes, I did. Calling it that does not make it so, however. And declaring that it's not doesn't remove the fact that it is. > > All naturals are a finite. Therefore your infinite number ...111 > cannot be a natural. QED. > If the "countably infinite" string is "finite but unbounded", then indeed it can and must be. And its successor is 0. Smiles, Tony |