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seen Aug 5 at 15:11

Jul
26
comment A question about sentences undecidable in Peano's Arithmetic
I am really trying to avoid theories in which second order axioms are "represented" by first order axiom schemes. These are actually first order theories.
Jul
26
comment A question about sentences undecidable in Peano's Arithmetic
I am really trying to avoid theories in which second order axioms are "represented
Jul
25
comment A question about sentences undecidable in Peano's Arithmetic
The question at the end of my previous comment is not quite what I meant to ask. I meant to ask whether every consistent and axiomatizable extension of PA-even if based upon higher order logic-has non-standard models which contain "infinite" positive integers. I am not too sure about exactly what Godel's incompleteness theorem and model theory have to say regarding consistent and axiomatizable extensions of PA that are not based upon first order logic.
Jul
24
comment A question about sentences undecidable in Peano's Arithmetic
Many thanks for your answers which have given me alot to think about. I thought that Hilbert and Bernays presented the first formalized and axiomatizable version of what I call SOA. You do not mention whether any of the interesting examples of sentences such as the Paris-Harrington theorem or Goodstein's theorem-which have been shown to be undecidable in PA-are also undecidable in SOA. Finally, does every axiomatizable theory-even if based upon higher order logic-have non-standard models which contain "infinite" positive integers?
Jul
24
accepted A question about sentences undecidable in Peano's Arithmetic
Jul
23
asked A question about sentences undecidable in Peano's Arithmetic
May
5
comment A question about simple closed curves in finite dimensional Euclidean spaces
Thanks for the clarification
May
3
comment A question about simple closed curves in finite dimensional Euclidean spaces
Many thanks for such a complete answer. I am not too familiar with homology theory so it may take me awhile to fully understand your proof.
May
3
accepted A question about simple closed curves in finite dimensional Euclidean spaces
May
2
asked A question about simple closed curves in finite dimensional Euclidean spaces
May
1
comment A question about Cantor's Power Set theorem without the Axiom of Choice
You are right. I was finally able to digest all the steps of your proof. Proving theorems about all infinite cardinal numbers can be quite tricky when the Axiom of Choice is not available.
Apr
30
comment A question about Cantor's Power Set theorem without the Axiom of Choice
In ZF set theory without the Axiom of Choice, there exist infinite sets X which are neither Alephs nor Dedekind-finite. Is it still true for such sets that CARD(2^X) is greater than 2*CARD(X)?
Apr
29
comment A question about Cantor's Power Set theorem without the Axiom of Choice
Thanks for the neat proof-in ZF without the Axiom of Choice-that CARD(X(I)) is greater than CARD(X) when X is infinite.
Apr
29
accepted A question about Cantor's Power Set theorem without the Axiom of Choice
Apr
28
asked A question about Cantor's Power Set theorem without the Axiom of Choice
Apr
4
awarded  Popular Question
Mar
6
awarded  Yearling
Mar
3
comment Questions about a possible way of representing construcive ordinal numbers
I was trying to work out whether any useful notation systems for a segment of the countable ordinal numbers could be developed, using recursive functions-or even just primitive recursive functions-as the notations. When you pointed out the existence of subsets of K which are densely ordered by "<", I saw what was wrong with this idea and why Hardy (who, I believe, first investigated the ordering "<") never went very far with it.
Mar
2
accepted Questions about a possible way of representing construcive ordinal numbers
Mar
1
comment Questions about a possible way of representing construcive ordinal numbers
Please disregard this last comment