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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.
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
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
1
vote
Accepted
Does mathematical induction presuppose the existence of a completed infinity?
This is quite a mouthful for a question but Peano Arithmetic does not seem to require infinity whereas Peano Axioms (second order) does seem to be equivalent to an axiom of infinity.
- 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
1
vote
Defining the standard model of PA so that a space alien could understand
Not sure about aliens but if we had to approach the task of explaining $\mathbb N$ to, say, the Pirahãs (who have presumably not been exposed to any modern mathematics), we would first have to answer …
- 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
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
2
votes
How is compactness related to countable saturation?
It turns out that there is a direct relationship between Cantor's lemma for nested compact sets $K_n$, on the one hand, and nested sequences of internal sets, on the other. Namely, the latter can be v …
- 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
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
4
votes
Are all countable, nonstandard models of arithmetic given by ultrapowers?
It could be mentioned perhaps that Skolem's non-standard model of arithmetic (1933-1934) is countable and is a kind of a "definable" version of the ultraproduct construction. Namely, Skolem only uses …
- 16.6k
2
votes
Non-ZFC set theory and nonuniqueness of the hyperreals: problem solved?
The usefulness of the hyperreals stems from such tools as saturation and the transfer principle. These tools are available in other superfields R' of R only to the extent that one can construct morph …
- 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
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
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