# Tag Info

## Hot answers tagged reverse-math

### Is most of mathematics independent of set theory?

EDIT: for a variety of reasons, I think I should give an explicit caveat here. Obviously I believe the things in my answer below - otherwise I wouldn't have written it. But I am sure there are many, ...
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### Is most of mathematics independent of set theory?

You can think of set theory as a low-level programming language, like Assembly; it works directly with sets in a way analogous to how Assembly works directly with bits and bytes. Some people work with ...
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### Is most of mathematics independent of set theory?

Most mathematics can be translated into some suitable set theory such as ZFC. That is certainly true! However, it is totally different from the claim that most mathematics deals with sets. From the ...
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### How are sets defined in reverse mathematics?

The answer is in the name of the book. The system described is equivalent to a particular (monadic) second-order theory using Henkin semantics. Instead of talking about "sets", Simpson could have ...
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### Is the set of all algorithms computable?

This depends on the definition of "algorithms". If as stated informally in Wikipedia (often used in introductory algorithm courses), Starting from an initial state and initial input (...
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### Are there non-standard counterexamples to the Fermat Last Theorem?

I am going to respond to two questions quoted below, which come from this comment. Here "TP" means "transfer principle". I've switched the order of the questions. ... why it does not prove that ...
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### Is it consistent for $P(\mathbb{N})$ to be present in a low layer of the Constructible Universe?

No: $L_\alpha$ is countable for any countable $\alpha$, so it cannot contain all of $P(\mathbb{N})$.
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### Constructive proof of the Banach-Alaouglu theorem

If $X$ is separable then the closed unit ball $B$ of $X^{*}$ is metrizable in the $weak^{*}$ topology. [ $d(f,g)=\sum\limits_{k=1}^{\infty} \frac 1 {2^{i}} \frac {|f(x_i)-g(x_i)|} {1+|f(x_i)-g(x_i)|}$ ...
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### Computable but Nonexistent Set

This is a common confusion faced when learning reverse mathematics. There are three different objects to consider here: The function $g$. The graph of $g$, $\{\langle n,g(n)\rangle: n\in\mathbb{N}\}$....
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### How are sets defined in reverse mathematics?

I'm not sure I've understood your question correctly, but let me take a stab at it: The short version is that we can develop RCA$_0$ (and any other theory, for that matter) entirely "autonomously,...
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### How to get an original function from the limit definition of a derivative?

The derivative of a function $f$ at a point $a$ is defined as $$f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}.$$ Setting $f(x)=e^x$ and $a=0$ this yields \frac{d}{dx}e^x\mid_0 = \lim_{h\to 0}\frac{e^{0+...
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### Real numbers cannot be constructed?

This seems to me like a semantic issue: Nowadays, after the advent of the absolute infinite in mathematics, the axiom of infinity has begun being conventionally interpreted to posit the existence of ...
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### Could we take certain results as axioms and prove the original axioms using our new ones?

Yes, this has been intensely studied in a number of contexts. We pick some very small set of axioms, basically all the uninteresting ones; we then look at what implications this "base theory" can ...
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