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All right math geeks, lay it on me. What the hell is aleph zero? Is this the right symbol for it: [itex]\aleph_0[/itex]?
- Warren
- Warren
Originally posted by chroot
All right math geeks, lay it on me. What the hell is aleph zero? Is this the right symbol for it: [itex]\aleph_0[/itex]?
Originally posted by chroot
All right math geeks, lay it on me. What the hell is aleph zero? Is this the right symbol for it: [itex]\aleph_0[/itex]?
- Warren
Originally posted by chroot
So... a set with cardinality [itex]\aleph_0[/itex] has countably many elements?
What is the cardinality of the set of real numbers? They are uncountably infinite, right?
Also, I've seen people use [itex]\aleph_0[/itex] like a number -- they'' even say stuff like [itex]2^{\aleph_0}[/itex]. This just doesn't make any sense to me. Is it a number? If not, what is it?
- Warren
Originally posted by chroot
So... a set with cardinality [itex]\aleph_0[/itex] has countably many elements?
What is the cardinality of the set of real numbers? They are uncountably infinite, right?
Also, I've seen people use [itex]\aleph_0[/itex] like a number -- they'' even say stuff like [itex]2^{\aleph_0}[/itex]. This just doesn't make any sense to me. Is it a number? If not, what is it?
- Warren
{
{(0,A),(1,A)}
{(0,A),(1,B)}
{(0,B),(1,A)}
{(0,B),(1,B)}
}
Originally posted by Hurkyl
Minor correction; [itex]A^B[/itex] is the set of all functions from [itex]B[/itex] to [itex]A[/itex], and exponentiation for cardinal numbers is defined as [itex]|A|^{|B|}= |A^B| [/itex].
Originally posted by chroot
Okay so [itex]\aleph_0 = | \mathbb{Z} | = | \mathbb{N} | = | \mathbb{Q} |[/itex] and [itex]\aleph_1 = | \mathbb{R} |[/itex].
Is it acceptable to say that [itex]\aleph_1 > \aleph_0[/itex]? Or that the cardinality of the reals is larger than the cardinality of the integers?
I'm not sure I understand where the [itex]\mathfrak{c}[/itex] came from if [itex]\aleph_1 \equiv \mathfrak{c}[/itex].
Is there an [itex]\aleph_2[/itex], ad infinitum? This all seems funny to me, that these cardinal numbers obey different sorts of rules than normal numbers. I haven't gotten my head around it yet.
- Warren
Originally posted by chroot
Is it acceptable to say that [itex]\aleph_1 > \aleph_0[/itex]?
I'm not sure I understand where the [itex]\mathfrak{c}[/itex] came from
Is there an [itex]\aleph_2[/itex], ad infinitum?
Originally posted by Hurkyl
the class of cardinal numbers is "too big" to fit in a set.
Can you expand a little on this?
*mumbles* mommy.. mommy.. make it stop.Originally posted by Hurkyl
Therefore the set of all cardinal numbers cannot exist.
Originally posted by NateTG
Consider this:
[tex]|2^A| > |A|[/tex]
is strict for
[tex]A \neq 0[/tex]
Let [tex]G[/tex] be a mapping [tex]A \rightarrow \{0,1\}^A [/tex]. Then for every [tex]a \in A[/tex], [tex]G(a)[/tex] is a function [tex]A \rightarrow \{0,1\}[/tex]
Now, construct [tex]f:A \rightarrow \{0,1\}[/tex] in the following way:
[tex]f(a)= 1 [/tex] if [tex]G(a)(a)=0[/tex] and [tex]0[/tex] otherwise.
Clearly [tex]f[/tex] is not in the range of [tex]G[/tex] since [tex]G(a)(a) \neq f(a) \forall a \in A[/tex]
Therefore there are no surjective mappings [tex]A \rightarrow \{0,1\}^A [/tex], and no bijections can exist.
Proving the other direction is easy:
[tex]G(a)(b)=1 \iff a=b[/tex] is an injective function.
This proves that there are 'infinitely large' infinities.
Yep - but this is the grown-up versionOriginally posted by master_coda
Isn't that just Cantor's diagonal method?
Originally posted by chroot
What the hell is aleph zero?
Originally posted by uart
Question ? Some people above are referring to [tex] \aleph_1[/tex] as being equal [tex] \aleph_0^{\aleph_0}[/tex]. But why isn't [tex] \aleph_1[/tex] equal to [tex] 2^{\aleph_0}[/tex], since I've seen it shown that this is the next cardinal greater than [tex] \aleph_0[/tex] ?
Originally posted by master_coda
It hasn't been shown that [itex]2^{\aleph_0}[/itex] is the next cardinal greater than [itex]\aleph_0[/itex]. It can't be shown - since [itex]2^{\aleph_0}=\mathfrak{c}[/itex], the idea that [itex]2^{\aleph_0}=\aleph_1[/itex] is just a restatement of the continuum hypothesis.
Originally posted by uart
But since [tex]2^{\aleph_0}[/tex] can't be put into a 1-1 relation with the natural numbers then doesn't that mean that [tex]2^{\aleph_0}[/tex] is larger than [tex]\aleph_0[/tex] ? And if that is the case then why do you need to go all the way to [tex]\aleph_0^{\aleph_0}[/tex] to find the next thing bigger when [tex]2^{\aleph_0}[/tex] is bigger already ?
You have a right to be confused! Except for uart'sBut since can't be put into a 1-1 relation with the natural numbers then doesn't that mean that is larger than [tex]\aleph_0^{\aleph_0}[/tex] ? And if that is the case then why do you need to go all the way to to find the next thing bigger when is bigger already ?
I have never seen anyone refer to [tex]\aleph_0^{\aleph_0}[/tex]!Some people above are referring to [tex]\aleph_0^{\aleph_0}[/tex] as being equal .
Originally posted by HallsofIvy
I have never seen anyone refer to [tex]\aleph_0^{\aleph_0}[/tex]!
Where did you get that? I was under the impression [itex]{2}^{\aleph_0}= \mathfrak{c}[/itex] and that the assertion that this was equal to [itex]{\aleph_1}[/itex] was the "continuum hypothesis" but I will say again that I have never seen [itex]{\aleph_0}^{\aleph_0}[/itex] beforeSince [itex]{\aleph_0}^{\aleph_0}=\mathfrak{c}[/itex]
Originally posted by HallsofIvy
Where did you get that? I was under the impression [itex]{2}^{\aleph_0}= \mathfrak{c}[/itex] and that the assertion that this was equal to [itex]{\aleph_1}[/itex] was the "continuum hypothesis" but I will say again that I have never seen [itex]{\aleph_0}^{\aleph_0}[/itex] before
Aleph Zero, also known as ℵ0, is the smallest infinite cardinal number in set theory. It represents the size of the set of all natural numbers (1, 2, 3, ...), which is also known as a countably infinite set.
Aleph Zero is significant because it helps us understand the concept of infinity and the size of different infinite sets. It is also used in various mathematical proofs and theories, such as Cantor's diagonal argument and the continuum hypothesis.
Aleph Zero is the smallest infinite cardinal number, but there are larger infinite numbers such as ℵ1, ℵ2, and so on. These numbers represent the size of larger infinite sets, such as the set of all real numbers or the set of all subsets of natural numbers.
No, Aleph Zero cannot be counted or measured in the traditional sense because it represents the size of an infinite set. It is a theoretical concept used in mathematics to understand the properties of infinite sets.
Aleph Zero has applications in various fields of mathematics, such as topology, set theory, and analysis. It is also used in computer science and theoretical physics. Additionally, the concept of infinity and Aleph Zero has philosophical and metaphysical implications.