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1
vote
2answers
468 views

Axiom of choice and convergence

Hi fellows, I was wondering. Is the axiom of choice used to show that $\mathbb{R}$ is complete? If yes, is there a way to construct monotonic bounded sequences that do not converge? Thanks in ...
0
votes
1answer
420 views

Maximum sum of 3 consecutive numbers in a permutation [closed]

Given that $X = \{0, 1, 2, ..., 7, 8, 9\}$, and $P$ is a permutation on $X$. Let $M(P)$ be the maximum sum of 3 consecutive elements. For example, if $P = (0, 2, 4, 1, 5, 7, 9, 3, 8, 6)$, then $M(P)$ ...
12
votes
8answers
2k views

Are there any mathematical objects that exist but have no concrete examples? [closed]

I am curious as to whether there exists a mathematical object in any field that can be proven to exist but has no concrete examples? I.e., something completely non-constructive. The closest example ...
5
votes
4answers
2k views

Does constructing non-measurable sets require the axiom of choice?

The classic example of a non-measurable set is described by wikipedia. However, this particular construction is reliant on the axiom of choice; in order to choose representatives of $\mathbb{R} ...
5
votes
3answers
727 views

Are all group monomorphisms regular, constructively?

By "constructive" I mean something that would go through in CZF for example. [added Oct 6] A sketch of a standard proof (such as referenced in comment below), which is almost constructive: Let H be a ...
11
votes
7answers
1k views

What happens when we print the digits of a real number?

Here are two well known facts, which put together leave me confused. First, it's well known that intuitionistic logic is the logic of constructive mathematics. From every intutionistic proof, you ...
4
votes
7answers
2k views

Are real numbers countable in constructive mathematics?

We are talking about ordinary reals in constructive mathematics. Let represent each real number by infinite converging series: $$r = [\;(a_0,b_0),(a_1,b_1),...,(a_i,b_i),...\;]$$ $$where\quad a_i ...
5
votes
2answers
2k views

Primitive element theorem without building field extensions

Is there are nice way to prove the primitive element theorem without using field extensions? The primitive element theorem says that if $x$ and $y$ are algebraic over $F$ and $y$ is separable over ...
10
votes
8answers
2k views

Direct construction of the integers

Is there a direct construction of the integers which does not involve taking any quotients? I am of course aware of the usual construction. I am also aware of the nice axiomatic characterization of ...
7
votes
2answers
814 views

Why do generic polynomials work in reality?

I understand that a generic $G$-polynomial $f(t_1,...,t_n)[X]$ over field $k$ has Galois group $G$ over $k(t_1,...,t_n)$. And basically any $G$ extension of $k$ should be generated by a realization of ...
7
votes
1answer
622 views

Some constructive versions of the Continuum Hypothesis are false. Are any true, or open?

Background In constructive set theory (say based on CZF) there are inequivalent ways of stating the continuum hypothesis. Some of them are easily if not trivially refutable with common anti-classical ...
6
votes
4answers
724 views

Can dependent sums be encoded as dependent products?

Please forgive any unorthodox notation or obvious errors here... I'm trying to get an intuition for dependently typed languages, so I'm starting out by seeing which analogies I can take from the ...
9
votes
3answers
1k views

Is set-induction relatively consistent?

One way to state the axiom of foundation is that the $\in$ relation on any transitive set is well-founded in the following sense: A relation $(X,\prec)$ is well-founded if for any subset $S\subseteq ...
9
votes
4answers
2k views

reduced ⊗ reduced = reduced; what about connected?

Several questions actually. All rings and algebras are supposed to be commutative and with $1$ here. (1) Let $k$ be a field, and let $A$ and $B$ be two $k$-algebras. I need a proof that if $A$ and ...
18
votes
11answers
2k views

Are there any good nonconstructive “existential metatheorems”?

Are there any good examples of theorems in reasonably expressive theories (like Peano arithmetic) for which it is substantially easier to prove (in a metatheory) that a proof exists than it is ...
9
votes
3answers
829 views

Intuitionistic Lowenheim-Skolem?

Is there a version of the Löwenheim-Skolem theorem in intuitionistic logic? I'm particularly interested in the "downward" form. The standard proof I know uses the Tarski-Vaught test for ...