1
vote
3answers
60 views

How to prove that a set is infinite iff it is Dedekind infinite?

I need to prove the following: A set $X$ is infinite if and only if it is equipotent to a proper subset of itself Here, $X$ is defined to be infinite if $|X|$ is not a non-negative integer or ...
3
votes
2answers
87 views

Is there any infinite quantity small enough to be affected by finite changes?

Hilbert's paradox of the Grand Hotel shows us, among other useful things, that the cardinality of any infinite set is a quantity equal to n more than itself for any finite n. I am interested in ...
-1
votes
2answers
163 views

Is Infinity Needed in Maths? Does Infinity Actually Exist? [closed]

I'm asking this question as I have been having an on going online debate with a friend of mine. I claimed that Infinity does in fact exist in Maths and in Reality, as there's a whole plethora of ...
3
votes
0answers
69 views

Is it possible for infinite sets to exist in ZFC with the negation of the Axiom of Infinity? [duplicate]

The Axiom of Infinity states that at least one inductive set exists. Inductive sets are infinite, but not all infinite sets are inductive. Suppose that we take ZFC with the negation of the Axiom of ...
0
votes
1answer
57 views

Newbie approach to understand generalized continuum hypothesis

There is this theorem that size of power set constructed from infinite set is "more" infinite than the previous set: $$ \begin{eqnarray*} \aleph_0 &= |\mathbb{N}| \\ \aleph_{n+1} &= ...
5
votes
2answers
74 views

How can I define $\mathbb{N}$ if I postulate existence of a Dedekind-infinite set rather than existence of an inductive set?

Suppose in the axioms of $\sf ZF$ we replaced the Axiom of infinity There exists an inductive set. with the Axiom of Dedekind-infinite set There exists a set equipollent with its proper ...
7
votes
1answer
210 views

New Axioms of Infinity

Axiom of Infinity says there is an inductive set (i.e. a set which includes $\emptyset$ and is closed under successor operator). Formally: $Inf:\exists x~(\emptyset\in x~\wedge~\forall y\in ...
0
votes
2answers
145 views

Is Continuum Hypothesis false? [closed]

The continuum hypothesis states that there is no set whose cardinality is strictly between that of the integers and the real numbers. However, it seems that it is possible to construct sets that ...
1
vote
3answers
168 views

Why are irrational numbers uncountable and rationals contable?

Question 1: Why are irrational numbers uncountable and rationals contable? I really struggle to understand this. I initially thought it had something to with the fact that between any two numbers ...
3
votes
1answer
135 views

Understanding countable ordinals (as trees, step by step)

Even though ordinal numbers – considered as transitive sets – are perfect non-trees, it is worth (and natural) to visualize them as trees, starting from the finite ones which are given as ...
6
votes
2answers
238 views

On the Continuum Hypothesis

Let me start out by saying that I am not a mathematician. I read an article over at Scientific American that discussed the Continuum Hypothesis. I developed the following thought experiment that would ...
1
vote
1answer
103 views

Aren't two infinite graphs always identical?

Suppose you have an infinite graph $G$. I assume $G$ to be cubic and planar. No further conditions, so it will be irregular, maybe in the sense of cubic planar version of Rado's graph: Every possible ...
2
votes
1answer
81 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
251 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 ...
6
votes
2answers
283 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
116 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}, ...
1
vote
1answer
95 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 ...
5
votes
4answers
252 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 ...
6
votes
1answer
338 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
109 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 ...
-1
votes
2answers
233 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
265 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, ...
8
votes
6answers
1k 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
180 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 ...
14
votes
1answer
526 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 ...
6
votes
6answers
476 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 ...
5
votes
1answer
795 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 ...
57
votes
6answers
3k 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. ...
11
votes
2answers
2k 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. ...