Tagged Questions
2
votes
1answer
63 views
Is there an element in $^* \Bbb N$ is Dedekind-infinite?
One definition of a finite set is that it can be injected into an initial segment of $ \Bbb N$, thus any $n$ in $\Bbb N$ is finite.
Accordingly, if it's legitmate to define every element in $^* \Bbb ...
4
votes
3answers
161 views
Why the need of Axiom of Countable Choice?
Two theorems:
$(1)$ Countable Union of Countable Sets is Countable
$(2)$ Cartesian Product of Countable Sets is Countable
Linked are the formal proofs on Proofwiki.
I do not understand why they ...
7
votes
2answers
157 views
Is the proper class of all ordinals equivalent to the potential infinity of pre-Cantor times?
My understanding is that the class of all ordinals is, by definition a proper class. This in the end is done to avoid a paradox: the collection of all sets would be paradoxical if you allow it to be a ...
1
vote
0answers
81 views
Relationship between ordinals and rank of well founded relations on $\mathbb N$
I want to understand the relation between ordinals and well founded relations on $\mathbb N$. I found a nice starting point here cut-the-knot/ordinals.
Ordinals start like this 0={}, 1={0}, 2={0,1}, ...
2
votes
2answers
62 views
Is there an infinite sequence AB, BC, CD, DX, …, YZ
Is it possible to construct an infinite set of ordered pairs of form S = {(A, B), (B, C), (C, D), (D, x), ..., (y, Z)}?
Every element (B, C...) must appear only once as the first object in one of the ...
1
vote
1answer
47 views
Finding an element of the intersection of an infinite sequence of “compatible” sets of infinite sequences
Let $A$ be a set. Let $A^\omega$ denote the set of infinite sequences of members of $A$ (i.e., functions from $\omega$ to $A$). Define $\omega_n = \omega \setminus \{n\}$. Let $A^\omega_n$ denote the ...
3
votes
4answers
168 views
Is the powerset of every Dedekind-finite set Dedekind-finite?
Is the powerset of every Dedekind-finite set Dedekind-finite?
I think this statement can be written in $\textbf{Set}$: If every mono (=injection) $f: A \to A$ is iso (=bijection), then every mono ...
1
vote
2answers
57 views
Prove that if a set is Peano finite, then it is Dedekind finite.
I understand that this should be done by induction, but I have very limited knowledge on proof by induction. Could someone explain it in a way which also makes clear exactly what each stage of ...
5
votes
1answer
136 views
Can an infinite cardinal number be a sum of two smaller cardinal number?
Let $\kappa$ be an infinite cardinal number.
My question is whether there are $\lambda$ and $\mu$ such that both $<\kappa$ but $\lambda+\mu=\kappa$?
If AC holds, then the answer is definitely ...
0
votes
1answer
92 views
Contour Infinites and Vector Spaces
We usually define Hilbert or finite dimensional vector spaces, and even topologies or differential geometry on $\mathbb{R}^n$ , so I wonder what is the implication of doing that on some extended ...
0
votes
2answers
183 views
How large is the infinity of real numbers [closed]
Umm ... Can someone disprove my proof that there are aleph-1 number of real numbers? Even comments to make my proof more rigorous are welcome.
https://www.dropbox.com/sh/1fz28jlwrprh4jv/rhA7Ad7OtX
2
votes
4answers
193 views
An infinite set having “one more element” than another infinite set
A classic example of homeomorphism is between a sphere missing one point and a plane
To see this, place a sphere on the plane so that the sphere is tangent to the plane. Given any point in the plane, ...
5
votes
6answers
714 views
Why accept the axiom of infinity?
According to my readings, Russell showed that a principle Frege used to reduce Peano arithmetic to logic lead to a contradiction. So, Russell tried to reduce mathematics to logic a different way but ...
2
votes
1answer
131 views
Can we get uncountable ordinal numbers through constructive method?
As we know, $2^{\aleph_0}$ is a limit ordinal number, however, it is greater than $\omega$, $\omega+\omega$, $\omega \cdot \omega$, $\omega^\omega$, $\omega\uparrow\uparrow\omega$, and even $\omega ...
12
votes
1answer
433 views
Is there an absolute notion of the infinite?
Skolem's paradox has been explained by the proposition that the notion of countability is not absolute in first-order logic. Intuitively, that makes sense to me, as a smaller model of ZFC might not be ...
1
vote
3answers
344 views
How to compare infinities
If A is the set of all real numbers in (0,1) with no '5' in the decimal representation, and B is the set with no '34' and '76446'. Then the set B is in some sense larger then A, how can I express this ...
5
votes
6answers
340 views
Book/article/tutorial as an introduction to Cardinality
I study CS, but on the first semester I have a lot of mathematics. Of course, there is an introduction to set theory and logic. Recently, we had lectures about cardinality, different kinds of ...
4
votes
1answer
445 views
Cardinality of a set that consists of all existing cardinalities
I have taken a look at the following topics:
number of infinite sets with different cardinalities
Cardinality of all cardinalities
Are there uncountably infinite orders of infinity?
Types of ...
40
votes
6answers
2k views
Why is $\omega$ the smallest $\infty$?
I am comfortable with the different sizes of infinities and Cantor's "diagonal argument" to prove that the set of all subsets of an infinite set has cardinality strictly greater than the set itself. ...
9
votes
2answers
1k views
Are there uncountably infinite orders of infinity?
Given a set $S$, one can easily find a set with greater cardinality -- just take the power set of $S$. In this way, one can construct a sequence of sets, each with greater cardinality than the last. ...
9
votes
6answers
2k views
Are there more rational numbers than integers?
I've been told that there are precisely the same number of rationals as there are of integers. The set of rationals is countably infinite, therefore every rational can be associated with a positive ...
8
votes
8answers
1k views
What is larger — the set of all positive even numbers, or the set of all positive integers?
We will call the set of all positive even numbers E and the set of all positive integers N.
At first glance, it seems obvious ...
