New Anti-Cantor Attack
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From: Aatu Koskensilta on 4 Jun 2010 11:54 David Bernier <david250(a)videotron.ca> writes: > Feferman also discusses ACA_0, a theory having to do with > arithmetic-sentence based analysis or something like that. ACA_0 is a conservative extension of PA in the language of second-order arithmetic -- that is, all of its theorems which can be expressed in the language of PA are provable in PA, and conversely. It has both number and set variables, and as axioms induction stated as a single sentence: For all sets X, if 0 is in X, and n+1 is in X whenever n is, then all numbers are in X. the usual axioms for successor, addition and multiplication, and a predicative comprehension schema: For any formula P containing no bound set variables, the universal closure of There exists a set X such that a number x is in X if and only if P(x). is an axiom. This theory is predicatively justified, in the sense that we can make predicativist sense of its axioms, and can provide arguments for them that are compelling on the predicativist conception of mathematics. (It is also finitely axiomatizable, unlike PA itself.) -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 4 Jun 2010 12:09 David Bernier <david250(a)videotron.ca> writes: > Poincare was an intuitionist Sure, but not in the sense the term is used today in foundations and mathematical logic. > Every natural number can be given a finite description, a numeral. > This is not so for a typical subset of N, P(N), P(P(N)), etc. Right, hence the qualms of various predicatively and vaguely constructively inclined mathematicians at the turn of the 20th century. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: MoeBlee on 4 Jun 2010 13:50 On Jun 2, 4:40 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > It amazes me how Clinton was ridiculed a decade ago for > asking for the definition of "is," He didn't ask for a definitionof 'is'. He correctly and legitimately pointed out that the answer to the question posed to him depends on which sense of the word 'is' was meant. MoeBlee
From: David R Tribble on 4 Jun 2010 17:39 |-|ercules wrote: > Here's my equivalent proof of uncountable infinity. > Let's assume an enumeration of naturals exists, call it N. > N= { 1, 2, 3, 4, ... } > > Let's calculate a new natural MAX+1. > That is 4+1 = 5 > in this finite subset example. > Voila 5 is a new number not in [this finite subset of] N > > Therefore no matter how big N is there is always a new element > that can be listed and therefore the size of the set N is bigger than infinity. Ah, but how much bigger than infinity? One more bigger? Twice as big? Infinity times bigger? Infinity squared bigger? Oh, and is MAX the same as the size of N, or is it something different? Details, man, details. It's the precise details that matter.
From: David Bernier on 5 Jun 2010 02:32
Aatu Koskensilta wrote: > David Bernier<david250(a)videotron.ca> writes: > >> Feferman also discusses ACA_0, a theory having to do with >> arithmetic-sentence based analysis or something like that. > > ACA_0 is a conservative extension of PA in the language of second-order > arithmetic -- that is, all of its theorems which can be expressed in the > language of PA are provable in PA, and conversely. It has both number > and set variables, and as axioms induction stated as a single sentence: > > For all sets X, if 0 is in X, and n+1 is in X whenever n is, then all > numbers are in X. > > the usual axioms for successor, addition and multiplication, and a > predicative comprehension schema: > > For any formula P containing no bound set variables, the universal > closure of > > There exists a set X such that a number x is in X if and only if P(x). > > is an axiom. > > This theory is predicatively justified, in the sense that we can make > predicativist sense of its axioms, and can provide arguments for them > that are compelling on the predicativist conception of mathematics. (It > is also finitely axiomatizable, unlike PA itself.) > Thanks for your explanation. Poincare wasn't a slouch, and neither were Weyl or Brouwer. So I try to grasp their point of view, to the extent possible. Admittedly, there are many existence proofs today which use the Axiom of Choice, and reals can't all be listed. The other side of the coin for me goes like this: I like to think (picturing the Cantor set), that any ternary expression 0.b_1 b_2 b_3 ... b_j ... 1<=j< oo , b_j in {0, 2} makes sense. Then there is measure theory according to which every countable subset X of the reals has Lebesgue measure zero. (And so on). All of Lebesgue measure theory in R^n, compactness and connectedness (general topology), Baire category, make sense to me. So I'm sort of disappointed with what I read of Poincare in that for me, it doesn't seem to give a fully satisfactory account or description of the continuum (the reals, the real number line): what is there to find in the real number line? What reals exist? And, BTW, if you happen to know what Weyl and/or Poincare thought of Dedekind cuts and/or Cauchy's development of Cauchy sequences, what with "every Cauchy sequence converges", I'd be happy to hear about it. David Bernier |