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Jul 26, 2015 at 18:31 comment added Garabed Gulbenkian I am really trying to avoid theories in which second order axioms are "represented
Jul 25, 2015 at 17:32 comment added Garabed Gulbenkian 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, 2015 at 18:09 comment added Garabed Gulbenkian 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, 2015 at 16:35 comment added Noah Schweber This is mostly true, but the last paragraph is wrong: TA has lots of nonstandard countable models.
Jul 24, 2015 at 12:46 review First posts
Jul 24, 2015 at 13:18
Jul 24, 2015 at 12:42 history answered none CC BY-SA 3.0