All Questions
Tagged with model-theory nonstandard-analysis
27
questions
5
votes
1
answer
193
views
What is the theory of computably saturated models of ZFC with an *externally well-founded* predicate?
For any model of $M$ of ZFC, we can extend it to a model $M_{ew}$ with an "externally well-founded" predicate $ew$. For $x \in M$, We say that $M_{ew} \vDash ew(x)$ when there is no infinite ...
- 6,298
6
votes
5
answers
2k
views
Standard models of N and R: An Alice/Bob approach
This is a question about a comment in a recent publication by Roman
Kossak. Kossak wrote:
"Nonstandardness in set theory has a different nature. In
arithmetic, there is one intended object of ...
- 15.8k
9
votes
1
answer
288
views
Transfinitely iterating the Levi-Civita, Hahn or Puiseux constructions
This question was originally asked at MSE but seems too advanced, so I'm reposting it here.
In short, the idea is that many constructions for non-Archimedean fields can naturally be iterated, in some ...
- 4,857
12
votes
2
answers
551
views
Decidability of a first-order theory of hyperreals
The theory of real closed fields is decidable. The hyperreals satisfy that theory, so we can interpret statements in the theory of real closed fields as being about hyperreals.
If we add a unary ...
- 6,298
6
votes
0
answers
196
views
Isomorphism of hyperreal fields viewed as extensions of the field of reals
I asked this question on Mathematics Stackexchange but got no answer.
Question. Does $ZFC$ prove that there are non-principal ultrafilters $\mathcal U$ and $\mathcal V$ over $\mathbb N$ such that the ...
- 4,376
9
votes
2
answers
442
views
Can nonstandard fields contain $\mathbb R$ in different ways?
Suppose $e : \mathbb R \to F$ is an elementary embedding in the language of ordered fields. Can there exist an elementary embedding $e' : \mathbb R \to F$ such that $e \not= e'$? Note that it would ...
- 18.1k
6
votes
1
answer
321
views
Cofinality of infinitesimals
Suppose $\kappa$ is an infinite cardinal and $U$ is a countably incomplete uniform ultrafilter over $\kappa$. Then $\mathbb R^\kappa/U$ is nonstandard. What is the cofinality of the set of ...
- 18.1k
2
votes
0
answers
170
views
How to construct "inaccessible hypernatural"?
Consider that, take a sufficient large natural number $a_1$, then take a natural number $a_2$ sufficient large to $a_1$, then take $a_3$,...
Now we have a function $n \mapsto a_n$ which grows very ...
- 475
4
votes
0
answers
144
views
Self homomorphisms of hyperreals fixing the reals
What do we know about the circumstances (whether having to do with the axioms of set theory or the model itself) under which a field $F$ of hyperreals (=ultrapower of $\mathbb R$ with respect to a non-...
- 2,383
8
votes
1
answer
518
views
What is the Turing degree associated with an ultrafilter $U$?
I asked Turing degree of a turing machine with access to an (arbitrary) nonstandard integer, not thinking about the possiblity that this could depend on the model used. The question was not formulated ...
- 6,298
5
votes
3
answers
824
views
Turing degree of a turing machine with access to an (arbitrary) nonstandard integer
Let us consider Turing machines (or other Turing-complete model of computation) that, in addition to their regular input, are given some integer $H$, where $H$ is positive nonstandard. This means, in ...
- 6,298
7
votes
1
answer
555
views
Are the definable hyper-reals, using quantifiers only over the standard reals and natural numbers, the same as the algebraic numbers?
This question arose today at Yevgeny Gordon's talk, "Will nonstandard analysis be
the analysis of the future?" at the CUNY Logic
Workshop. Here is my way of asking it.
Consider the ordered real field ...
- 230k
20
votes
2
answers
977
views
Automorphisms of the hyperreals over the rationals and nontrivial automorphism groups
A classic result says the automorphism group of $\mathbb{R}$ (over $\mathbb{Q}$) is trivial. The proof is simple: every automorphism preserves squares, and hence fixes the positive reals, so it must ...
- 201
2
votes
1
answer
310
views
Is there a model of ZF+ACC where transfer fails for the definable hyperreals?
In 2003 Kanovei and Shelah constructed a definable hyperreal field. The ultrapower used exploits a fairly large index set so that it is clear that the usual proof of Los and transfer does not go ...
- 15.8k
3
votes
1
answer
1k
views
What is the modern consensus on the difficulty of infinitesimals?
At a related thread at MSE an expert in reverse mathematics noted that "As the modern consensus is that only nonstandard models have infinitesimals, it will be quite challenging to give a concrete ...
- 15.8k