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Results tagged with lo.logic
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user 28128
first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
23
votes
A remark of Connes on non-standard analysis
Connes' critique was recently analyzed by Kanovei, Katz, and Mormann in this article in Foundations of Science (see also arXiv 1211.0244). Here is the abstract:
We examine some of Connes' criticisms …
- 16.6k
15
votes
Set theoretical multiverse and truths
There is a subtle issue here but it is not where the OP thinks it is. Any explicitly written integer is obviously "standard" whereas each new integer arising in the ultrapower of $\mathbb N$ is obviou …
- 16.6k
12
votes
What is... a grossone?
I would like to summarize some findings concerning the mathematics of Sergeyev's grossone.
(1) Sergeyev's writing seems to contain confusion between the notions of ordinal
and a cardinal numbers. Th …
- 16.6k
10
votes
Circular, or missing, definition in set theory?
Thinking about the distinction between language and metalanguage may be helpful here. When one describes set theory as possessing a single binary relation denoted $\in$, one is operating at the level …
- 16.6k
8
votes
A remark of Connes on non-standard analysis
A recent article by Leichtnam and myself (arxiv) in the American Mathematical Monthly contains a "theorem" to the effect that, in the presence of a construction of the hyperreals, the following is tru …
- 16.6k
8
votes
Nonstandard analysis in probability theory
The answers given earlier are excellent. I would merely like to supplement them by the observation that the success of NSA and IST in probability and related fields is attested to by the fact that ne …
- 16.6k
7
votes
Can infinity shorten proofs a lot?
I once had a teaching assistant in calculus who admitted that he was unable to give an epsilon, delta proof that a the Heaviside stepfunction was not continuous. To take a slightly less trivial examp …
- 16.6k
7
votes
Was the early calculus inconsistent?
I would agree with Alexandre Eremenko's answer. The early calculus in fact was not inconsistent, as elaborated below.
Joël's answer is based on a premise that "the question is not precise enough to …
- 16.6k
7
votes
Is GCH useful in proving theorems?
Dixmier traces are
easily constructed in ZFC and there is an extensive literature on the
topic. Connes pointed out that such a trace with particularly good
properties can be constructed in the assump …
- 16.6k
6
votes
Undefinability of $\mathbb{Z}$ in the reals
The theory of real closed fields is complete and if the integers were definable in $\mathbb R$ this would contradict Goedel's incompleteness result.
- 16.6k
5
votes
What's wrong with the surreals?
A quick search indicates that Peano axioms are not mentioned on this page. It seems reasonable to mention that there does not seem to be a good notion of natural number in the surreals that would sati …
- 16.6k
5
votes
Accepted
Non standard extension of real numbers via nonprincipal ultra filters
As far as I understand the question is still open under $\neg CH$, namely whether isomorphism of hyperreal fields implies equivalence of filters (up to permutation of index set). Perhaps one can try t …
- 16.6k
5
votes
Real reverse mathematics
Here an interesting case study concerns the case $M=$ Leibniz. We have undertaken some detailed studies of primary documents recently, resulting in publications in the British Journal for the History …
- 16.6k
5
votes
Propositions equivalent to the completeness of the real numbers
Let $R$ be an Archimedean ordered field, and $S$ a non-trivial ultrapower extension of $R$. Then $R$ is complete if and only if $S$ admits a standard part; namely, every limited element of $S$ is inf …
- 16.6k
4
votes
A remark of Connes on non-standard analysis
Specifically with regard to the issue of what Connes means when he says that you can't "name" an infinitesimal: this issue was discussed above on this page, and I think we provided an answer that's di …
- 16.6k