All Questions
Tagged with real-numbers set-theory
79
questions
-2
votes
1
answer
573
views
is a set of equivalence classes in R x R countable? [closed]
An equivalence class in R x R is uncountable but is the set of equivalence classes in R x R countable?
- 15
-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
3
answers
371
views
Are there real numbers that cannot be uniquely expressed with a finite number of symbols?
Are there real numbers that cannot be uniquely expressed with a finite number of symbols? Is this the same thing as an uncomputable number?
I can show that if there is one such number then there are ...
- 1,894
2
votes
1
answer
645
views
Why set of real numbers not a set of ordered pairs?
Why set of real numbers not a set of ordered pairs ?
We write $\mathbb{R}^2 = \mathbb{R} \times \mathbb{R}$, then we define addition and multiplication on this new set. Together with those ...
user312097
4
votes
1
answer
153
views
Family of almost disjoint subsets of $\mathbb{R}_+$ (where non-disjointness is bounded)
Let $\mathbb{R}_+ = [0, \infty)$. I'm looking for a family $\mathcal{U}$ of $2^\mathfrak{c}$ subsets of $\mathbb{R}_+$, such that any member of $\mathcal{U}$ is unbounded in $\mathbb{R}_+$, but the ...
- 7,674
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
6
votes
2
answers
729
views
Well-orderings of $\mathbb R$ without Choice
The question is about well-ordering $\mathbb R$ in ZF. Without the Axiom of Choice (AC) there exists a set that is not well-ordered. This could occur two ways: a) there are models of ZF in which $\...
- 3,009
2
votes
1
answer
100
views
Decimal expansions and topological connectedness
I'm a bit confused by the following footnote from Moschovakis's Notes on Set Theory, p. 135fn24 (in the note, $\mathcal{N}$ denotes the Baire space). The puzzling part is in bold:
One may think of $...
- 5,537
1
vote
0
answers
111
views
Is there a model of ZFC in which every real number is definable? [duplicate]
https://en.wikipedia.org/wiki/Definable_real_number
Wikipedia defines a "definable real number" as one definable by a parameter-free formula in the language of set theory.
The article says, "The set ...
- 864
1
vote
1
answer
95
views
Fields of intermediate cardinality (2)
Assuming the existence of a cardinal $\aleph_0 <\mathfrak{m} < 2^{\aleph_0}$, does it follow that there is a subfield of $\mathbb{R}$ of cardinality $\mathfrak{m}$ containing no algebraic ...
- 169
2
votes
2
answers
178
views
Fields of intermediate cardinality
Assuming the existence of a cardinal $\aleph_0 <\mathfrak{m} < 2^{\aleph_0}$, does it follow that there is a subfield of $\mathbb{R}$ of cardinality $\mathfrak{m}$?
- 169
1
vote
2
answers
422
views
Real numbers for beginners
I am thinking about the Wikipedia (I understand disputed) article about “definable real numbers”.
It begins to say that,
A real number $a$ is first-order definable in the language of set theory, ...
- 1,817
11
votes
1
answer
1k
views
More than the real numbers: hyperreals, superreals, surreals ...?
I've read something about extensions of the real numbers, as hyperreals, superreals, surreals and, as I can understand, all these extensions contain some new kinds of infinitesimal and infinite ''...
- 63.1k
4
votes
1
answer
2k
views
Definable real numbers
Reading this Wikipedia page I found this definition:
A real number $a$ is first-order definable in the language of set
theory, without parameters, if there is a formula $\phi$ in the
language ...
- 63.1k
3
votes
1
answer
1k
views
Undefinable Real Numbers
Disclamer: I'm sure my definition of "definable" may be different than the/a established mathematical one, I am more than interested in learning why/how this is so, but that is not my question
Part 1:...
- 133
4
votes
1
answer
369
views
Maximal model for ℝ?
I have not dealt professionally with set theory, so excuse me if my way of formulating this question does not completely follow standard terminology. Actually, my question is about whether or not the ...
- 2,796
0
votes
1
answer
195
views
Axiom of choice and an example of a Well-ordered $\Bbb R$
From the axiom of choice we get that every set can be ordered in a way that will make it a well ordered set, including $\Bbb R$. However, since the ordinal of such a well-ordered set of $\Bbb R$ will ...
- 392
9
votes
4
answers
2k
views
Using Zorn's lemma show that $\mathbb R^+$ is the disjoint union of two sets closed under addition.
Let $\Bbb R^+$ be the set of positive real numbers. Use Zorn's Lemma to show that $\Bbb R^+$ is the union of two disjoint, non-empty subsets, each closed under addition.
- 517
11
votes
1
answer
2k
views
The well ordering principle
Here is the statement of The Well Ordering Principle: If $A$ is a nonempty set, then there exists a linear ordering of A such that the set is well ordered.
In the book, it says that the chief ...
- 563