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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
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
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
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
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
1
vote
What are the advantages of the more abstract approaches to nonstandard analysis?
One advantage of the more abstract axiomatic approach to nonstandard analysis was not mentioned when this page was active seven years ago, because the relevant mathematics was not yet available. As Ka …
- 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
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
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
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
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
3
votes
Euler's mathematics in terms of modern theories?
A somewhat delayed response is provided in our detailed study of Euler accepted for publication in Journal for General Philosophy of Science.
We apply Benacerraf's distinction between mathematical on …
- 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
2
votes
What is the modern consensus on the difficulty of infinitesimals?
The difficulty involved in developing a theory of infinitesimals that would be useful in analysis is illustrated by the fact that, as discussed in the comments above, the surreals are unsuitable for t …
- 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
4
votes
What is... a grossone?
My original question focused on Professor Lolli's apparent advocacy of Sergeyev's work. Thanks to the efforts of several editors the picture with Lolli's article has become clearer to me, and I would …
- 16.6k