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visits | member for | 5 years, 6 months |
seen | Aug 5 at 15:11 | |
stats | profile views | 1,679 |
Jul
26 |
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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 |
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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 |
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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 |
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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 |
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A question about simple closed curves in finite dimensional Euclidean spaces
Thanks for the clarification |
May
3 |
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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 |
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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 |
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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 |
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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 |
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Questions about a possible way of representing construcive ordinal numbers
Please disregard this last comment |