# Questions tagged [nonstandard-models]

For questions specifically concerning models of arithmetic (which could be Peano Arithmetic, the first-order theory of the natural numbers, or some other system) which differs from the standard model by the existence of nonstandard numbers.

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### Can theories with axioms beyond arithmetic make false promises about integer existence?

In this paper, author Nik Weaver warns that there could be questions of $\Sigma_1$-validity of ${\mathrm{ZFC}}$ set theory. As I understand it, he suggests that the axioms of a set theory might be ...
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### Is there a least standard model of Peano Arithmetics?

Here's the informal part: I'm trying to get a better intuition about the differences between the standard model and the non-standard ones. Some people seem to be really good at imagining what non-...
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### Totally order turing machines by halting

Does there exist some computable relation "<" on turing machines, such that the set of machines that halt is an initial segment. (If A and B are Turing machines, and A halts and B doesn't ...
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### System in the state space with differentiating links in feedback

My question is a copy of the following topic, but slightly reformulated: Representation in state-space of a system with ideal and real differentiating links in feedback There are two systems in the ...
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### No single axiom stating non-Archimedeanity [duplicate]

An ordered field $K$ with the ordering $<$ is Archimedean if for any $x \in K$ there exists $n \in \mathbb{N}$ (where $\mathbb{N}$ is the copy of the natural numbers in $K$) such that $|x|<n$. ...
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### On an explicit model for $\mathbf Q$ with the negation of the axiom of induction

I've been thinking for a while about non standard models of Robinson Arithmetic (here on out referred to as $\mathbf Q$), specifically ones in which induction ($\mathbf{AI}$) fails. This could happen ...
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### In what sense formal proof is a finite set of formulas?

The question is kinda vague I guess, but I'll try to explain. Suppose we have a formal system $\mathcal{FS}=\left(\Sigma, F, A, \Gamma\right)$, $\Sigma$ is an alphabet, $F\subset\Sigma^*$ is a set of ...
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### Strongest tools to detect unsoundness

I am curious about the strongest methods we currently know of that can allow us to discover (arithmetical) unsoundness or nonstandardness in a foundational system $S$ that interprets at least ACA (...
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### Uniformly Bounded Functions of Hyperreals

I am trying to study ultraproducts and non-standard analysis, and I have the following question. Fix a non-principal ultrafilter $\mathcal{F}$ to construct the hyperreal field ${}^*\mathbb{R}$. An ...
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### Is There a notion of Convergence for the Non-Standard Reals? How can we do Analysis?

I was reading up on Non-Standard Analysis from an Analytical perspective, i.e., how to do Analysis when I got stuck thinking of how we can do Analysis without even having a "traditional" metric , e.g.,...
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### Any set can be easily replaced by a base set with the same size

Article Forcing in nonstandard analysis states in the page 266: "For an infinite ordinal $\alpha$, any set $X$ such that every element of an element of $X$ has rank $\alpha$ is a base set and hence ...
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### Internal Set Theory: Start with nonstandard finite E, then the standard subset of E is infinite?

I'm working on understanding Edward Nelson's Internal Set Theory. I wonder if folks can check my reasoning. I need to reference a result which says that if E is a set then the following two ...
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### Elementary equivalence of standard and non-standard model of arithmetic

There is the common construction of a non-standard model of arithmetic by adding a constant symbol c to the signature and adding $\{n<c:n\in \mathbb{N}\}$ to the theory PA. Now by adding all ...
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### Generalizing the construction of $PA^\omega$

I'm taking an extremely basic course in mathematical logic where we briefly talked about nonstandard models of arithmetic. The only example we worked through has been the construction of $PA^\omega$ ...
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### Exponentiation and a weak fragment of arithmetic

The title is slightly misleading, since the theory I'm looking at is an extension of PA; however, the question is "morally" about very weak arithmetic. Specifically, consider the following theory PA':...
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### Is the standard model for the language of number theory elementarily equivalent to one with a nonstandard element?

On page 89 in A Friendly Introduction to Mathematical Logic, the author writes that the standard model $\mathfrak{N}$ for $\mathcal{L}_{NT}$ is elementarily equivalent to a model $\mathfrak{A}$ that ...
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### The satisfaction relation is undefinable, but does it still “exist”?

In Jech's book "Set Theory" chapter 13, he shows how the satisfaction relation $\models_{n}$ for $\Sigma_n$ formulas can be formalized in ZF. As he pointed out previously, the full satisfaction ...
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### Is the union of non-standard analogs of a family of sets a proper subset of the non-standard analog of the union of those sets?

The book on formal logic I'm using for self-study builds the foundation for non-standard analysis (to illustrate the usage of non-standard models) using the following construction. Consider the real ...
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### Is there a combined theory of the Reals and the Naturals that has a model where the Naturals and Reals have the same cardinality

The Upward Lowenheim-Skolem theory decrees that there must be a (non-standard) model of the naturals of cardinality the same as that of the standard model of the Reals. For any combined theory of the ...
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### Non-standard integers?

Edit: Realized almost immediately that this was a stupid question - see comment below. Was about to delete it when an answer appeared that seems like saving... Context: I just saw a presentation of ...
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### Having Trouble Seeing Why Friedman's Theorem (1973) is true.

I am reading about non-standard models of peano arithmetic, and came across a theorem by Friedman that states the following. Every non-standard countable model of (peano) arithmetic is isomorphic ...
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### Any non-standard halting oracles stronger than $\mathbb N$?

Let $M$ be some model of PA. Let $H_M$ be the set of codes of standard turing machines $X$ such that $M \models X \text{ halts}$. For example, $H_\mathbb N$ corresponds to the regular halting oracle (...
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### Non-standard model of arithmetic - why is adding new constants to the model acceptable?

I have a question about the following proof of existence of a model of non-standard arithmetic (taken from wikipedia): The existence of non-standard models of arithmetic can be demonstrated by an ...
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### Relationship between nonstandard analysis and nonstandard models of arithmetic

So I've been reading through some introductory texts on nonstandard analysis, basically through the ultrafilter construction of hyperreals and the transfer principle. It seems that much of the time, ...
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### Model of concatenation theory with left-cancellation but no right-cancellation

The theory of concatenation (TC) can be equivalently expressed as the following assumptions: Closure of strings under concatenation $+$. Existence of an empty string $e$, namely $e+x = x = x+e$ for ...
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### Information about $ZFC + (\neg CH)$

$\underline{\text{Some Background}}$: A classic axiomatic system of set theory in modernity is $ZFC$. Results from modern mathematical logic guarantee the existence of statements undecidable given ...
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### What do non-standard cardinalities look like?

The Von Neumann universe $V$ satisfies ZFC, and there are other models within $V$ that are non-standard and satisfy ZFC. If we look at one of these non-standard models $M$ with a non-standard model of ...
Let $\hat{V}$ be the standard universe constructed as $\hat{V} = V_0\cup V_1 \cup V_2 \cup ....$ where $V_0$ is the set of primary elements and $V_{v+1} := V_v \cup P(V_v)$ For the following (...