All Questions
Tagged with real-numbers set-theory
79
questions
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101
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Skolem's Paradox and undefinable reals
I'm trying to understand Skolem's paradox, and also some related ideas about definable numbers. I'm pretty new to learning about model theory. I'll lay out what I think I'm understanding, and maybe ...
- 293
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31
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Set representation of (real) numbers [duplicate]
Using the Von Neumann representation we can represent the non-negative whole numbers using the empty set, e.g. $1$ as {$\emptyset$}.
How do we represent with this notation numbers like $\sqrt2, -1, \...
- 43
3
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1
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116
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In ZF, is it possible that there is no cardinal such that Reals injects into?
Working in ZF, is it possible that there is no cardinal number such that $\mathbb{R}$ can inject into? For if there exists a cardinal number $\kappa$ such that $\mathbb{R}$ injects into $\kappa$, then ...
- 249k
0
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105
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Can every perfect set that is not a closed interval, or $(-\infty,a]$, or $[b,\infty)$ be written as a union of these types of intervals?
I have been reading the book "Introduction to Set Theory" by Jech and Hrbacek and have come to the following exercise in the chapter on sets of real numbers :
Every perfect set is either an ...
- 595
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4
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2k
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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.
- 11.5k
3
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1
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216
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Can you prove that $\Bbb R$ is uncountable using the Lebesgue measure?
I have been studying measure theory from the ground up, and am quite excited by the seeming power it holds. I thought of this last evening, and I wish to ask if the following proof of uncountability ...
- 10.4k
11
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1
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2k
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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 ...
- 3,047
4
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3
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450
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Possible interpretation of real numbers as functions? [duplicate]
At the end of the day, a real number can be viewed simply as a function over the integers —> the naturals which tells you the digit as that ten’s place (assuming base ten)? You could augment this ...
- 4,655
1
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1
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144
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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 ...
- 249k
1
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2
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167
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Continuum family of continuum subsets of $\mathbb R$ which are not pairwise order isomorphic
I need to construct a continuum family of pairwise order inequivalent subsets of $\mathbb R$, such that cardinality of intersection of every constructed subset and every nontrivial interval is also ...
- 620k
11
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1
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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 ''...
- 5,125
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63
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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?
- 397k
3
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74
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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
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138
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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
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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 $\beta$, ...
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