All Questions
16
questions
1
vote
2
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96
views
why is $L=\{\{x\mid x<q(i)\}\mid i\in\mathbb{N}\}$ not the set of all Dedekind cuts?
Let the set $L$ be definded as
$$L=\{\{x\mid x<q(i)\}\mid i\in\mathbb{N}\},$$
where $q(i)$ is some bijection from $\mathbb{N}$ to $\mathbb{Q}$.
Clearly, every member of $L$ is neither an empty set ...
- 1,266
0
votes
0
answers
92
views
Is axiom of choice required to throw away repeated intervals in a constructive argument?
I am looking at this answer to this question:
Let $U \subseteq \mathbb{R}$ be open and let $x \in U$. Either $x$ is rational or irrational. If $x$ is rational, define
\begin{align}I_x = \bigcup\...
- 743
1
vote
1
answer
144
views
Sets with Unique Subset Summing to Every Real
Do there exists sets of reals such that every real has a unique subset that sums to it. Formally, do there exists sets $S\subset\mathbb{R}$ such that every $r\in\mathbb{R}$ has a unique (up to ...
- 2,346
5
votes
1
answer
258
views
Contradiction of axioms of real numbers
I am just starting out in real analysis, so please bare with me. My questions concerns three specific properties of the real numbers, at least as far as i understand them. Those are:
The natural ...
- 71
1
vote
0
answers
90
views
Kolmogorov's construction of real numbers cardinality of functions that represent real numbers
Hi i am reading about lesser know construction of real numbers by Kolmogorov. In his construction real numbers are defined as a set $\Phi$ of functions $\alpha: \mathbb{N} \rightarrow \mathbb{N}$ that ...
- 11
-5
votes
1
answer
148
views
The set of irrationals numbers is countable?
I tried to prove this using statement using the difference of sets
$\mathbb{R}-\mathbb{Q}$ and the fact that $\mathbb{R}$ is not countable and $\mathbb{Q}$ is countable.
In general, is it possible to ...
- 605
1
vote
1
answer
70
views
Can we uniquely define for arbitrary, real-valued, finite sequence $X$, infinitely many pairs (real-valued $f(X)$, rank order of elements of $f(X)$)?
For an arbitrary sequence $X$ of $n$ distinct real numbers, can we uniquely and exhaustively define a set of infinitely many pairs of the form: $[f_{j},$ order$(f_{j}(x))]$, where $f_{j}$ is a real-...
- 171
1
vote
1
answer
143
views
Defining real numbers to exclude incomputable numbers
The real numbers are normally constructed via Dedekind cuts or similar approaches, which result in incomputable numbers: numbers that no finite algorithm can produce to arbitrary precision. This is ...
- 4,450
0
votes
1
answer
63
views
Exist strictly increasing $f \colon \omega _1 \mapsto \mathbb{R}$ [duplicate]
Does there exist a strictly increasing injective function $f \colon \omega _1 \mapsto \mathbb{R}$, where $\omega _1$ denotes the first uncountable ordinal?
- 43
3
votes
0
answers
74
views
How high in the constructible hierarchy do you need to go to see Dedekind-incompleteness?
This is a follow-up to my questions here and here. Let $X= (A,+,*,<)$ be an ordered field. Let us define a constructible hierarchy relative to $X$ as follows. Let $D_0(X)=A\cup A^2 \cup \{+,*,&...
- 10.3k
5
votes
0
answers
138
views
Where is the first gap in the constructible hierarchy relative to a real closed field?
This is a follow-up to my question here. Let $X= (A,+,*)$ be a real closed field. Let us define a constructible hierarchy relative to $X$ as follows. Let $D_0(X)=A\cup A^2 \cup \{+,*\}$. For any ...
- 10.3k
2
votes
1
answer
105
views
Where is the copy of $\mathbb{N}$ in the constructible hierarchy relative to a real closed field?
Let $X$ be a real closed field. Let us define a constructible hierarchy relative to $X$ is defined as follows. (This is slightly nonstandard terminology.). Let $L_0(X)=X$. For any ordinal $\beta$, ...
- 10.3k
4
votes
1
answer
299
views
Does calculus need choice axioms?
To do calculus, we (presumably) need the real numbers (or perhaps some abstract complete metric space?).
When the real numbers are constructed using Cauchy sequences or Dedekind cuts, does this ...
- 2,106
-2
votes
1
answer
343
views
countability of a set of uncountable many real intervals?
Consider a real line $R$ and for each $i ∈ R$ there is a real line, $R_i$, that intersects $R$ at $i$; the $R$ and $R_i$ share only the number $i$ and let none of $R_i$ intersect. Since each $R_i$ ...
- 15
1
vote
0
answers
140
views
Brief overview of the foundations of math?
I am very interested in finding out about the foundations of math. When I feel foolish enough to try, I visit Wikipedia and end up in a truly endless rabbit hole. There are dozens of terms that I don'...
- 23.8k