Venn diagram for the reals and transfinite numbers as sets

  • #1
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My statement:
The first transfinite ordinal, omega is the first number that cannot be expressed by any natural number, therefore it is not included in the set of natural numbers. The set of natural numbers is a subset of real numbers, every natural number can be taken out of it, but still true that there is no integer number in it that is capable ("big enough") to pair with omega. By this, it is also a good statement to say that the set of transfinite ordinals and the real numbers are disjoint sets.

Is this a good tought? If this is not, then can be refined to make it mean that "omega is after the finite numbers therefore it is after also any real number"?
Please help me to make this statement more formal.
Is the Venn diagram below correct?

Thank you!
 

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  • #2
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My statement:
The first transfinite ordinal, omega is the first number that cannot be expressed by any natural number, therefore it is not included in the set of natural numbers. The set of natural numbers is a subset of real numbers, every natural number can be taken out of it, but still true that there is no integer number in it that is capable ("big enough") to pair with omega. By this, it is also a good statement to say that the set of transfinite ordinals and the real numbers are disjoint sets.

Is this a good tought? If this is not, then can be refined to make it mean that "omega is after the finite numbers therefore it is after also any real number"?
Please help me to make this statement more formal.
Is the Venn diagram below correct?

Thank you!
First of all, there is a difference between ##\omega##, the first infinite ordinal, and ##\Omega## the first uncountable ordinal.

Anyway, you are correct that the only thing that ##\mathbb{R}## and the ordinals have in common is ##\mathbb{N}##. They are not quite disjoint.

If you want to make precise that ##\omega## is bigger than any real, you'll need the surreal numbers.
 
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  • #3
nomadreid
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It seems a bit of a mix. First, I do not recall seeing upper-case omega Ω used for either the first transfinite ordinal (which is usually lower-case omega ω) or the first transfinite cardinal (which is aleph-zero ); nor have I seen it used for the first uncountable ordinal (ω1) or the first uncountable cardinal (Aleph-1), nor the cardinal corresponding to the real numbers (Which is 2aleph-zero I have seen Ω as the class of all ordinal numbers. "Number" is a bit vague. In any case, one cannot automatically associate the first uncountable ordinal with the The next term you could make more precise is "after". For ordinals, "bigger than" corresponds to set membership, but for size, cardinalities are usually compared. Your idea that "ω is after the finite numbers" is vague, but I would presume that this means that every finite ordinal is a member of ω. Even vaguer is "it [ω] is after also any real number". Since you are attacking this in terms of sets, you will need a good definition of a real number as a set. I am not sure how you intend to make sense of that statement. As far as your Venn diagram: if we stretch a point and take the set of real numbers to be the set of all subsets of the natural numbers, then yes, ℕ⊂ℝ. ω and ℕ could be argued to be the same unless you are a purist. Since I don't know what you mean by Ω, I do not know where that should go in your diagram.
My feeling is that you want to use some sort of transitivity, saying that saying that "a real number r is less than a finite natural number n which is less than ω , therefore r is less than ω." But this doesn't work: the first "less than" is according to a metric, the second one is set membership. So they are two different relationships, so transitivity doesn't apply.
 
  • #4
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It seems a bit of a mix. First, I do not recall seeing upper-case omega Ω used for either the first transfinite ordinal (which is usually lower-case omega ω) or the first transfinite cardinal (which is aleph-zero ); nor have I seen it used for the first uncountable ordinal (ω1) or the first uncountable cardinal (Aleph-1), nor the cardinal corresponding to the real numbers (Which is 2aleph-zero I have seen Ω as the class of all ordinal numbers.
https://en.wikipedia.org/wiki/Aleph_number#Aleph-one
Using ##\Omega## to denote all ordinals is something I have never seen before.
 
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  • #5
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It seems a bit of a mix. First, I do not recall seeing upper-case omega Ω used for either the first transfinite ordinal (which is usually lower-case omega ω) or the first transfinite cardinal ...
I used upper case omega, because I tried to symbolize the second part of this set, in which the lower case omega is only a member:
##0,1,2,\cdots, \omega , \omega +1, \omega + 2, \cdots , \omega+\omega, \omega + \omega + 1, \cdots##
So the upper case omega represents the ordinals that can not be represented by natural numbers. Maybe this is not a good idea, I'd be happy to follow your suggestions.

if we stretch a point and take the set of real numbers to be the set of all subsets of the natural numbers, then yes, ℕ⊂ℝ.
The set of all subsets of the natural numbers would be something like this, right? : { {0},{0,1},{0,1,2},...{0,2},{0,2,3}... } -- all of the possible combinations of the natural numbers (set of 2aleph0, right?) If you mean this, then no, not this is what I'd like to explain.
I simply mean the simple meaning of ℕ⊂ℝ, like this: http://thinkzone.wlonk.com/Numbers/RealSet-big.png

Yes, I try to mix the the ordinals and the cardinals to show that if a natural number is not big enough to represent ##\omega## in a well-ordered list, it is also true for its container set, the Reals.

Is this a defendable concept?
Is it possible to demonstrate this in a Venn diagram?
 
  • #6
nomadreid
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https://en.wikipedia.org/wiki/Aleph_number#Aleph-one
Using ##\Omega## to denote all ordinals is something I have never seen before.
You're right, micromass, for calling me out on the carpet on this. I was thinking, "I have seen Ω used somewhere for the class of all ordinals in a sum, but it is not standard ("On" is pretty standard, and "Ord(α)" is standard for saying that α is an ordinal), and I would have difficulty digging up the reference. Probably someone introducing their own notation, so it should be avoided.", and ill-advisedly wrote the first part of my thought. o:)
 
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  • #7
nomadreid
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I used upper case omega, because I tried to symbolize the second part of this set, in which the lower case omega is only a member:
##0,1,2,\cdots, \omega , \omega +1, \omega + 2, \cdots , \omega+\omega, \omega + \omega + 1, \cdots##
A minor note about style: your list looks deceptive, since you look like you are listing only a countable number of ordinals, not even getting up to
ω⋅ω. But OK,
So the upper case omega represents the ordinals that can not be represented by natural numbers.
In other words, On minus ω. This is not standard, and so should be explicitly defined before using it.
The set of all subsets of the natural numbers would be something like this, right?
Yes

if a natural number is not big enough to represent ω" style="font-size: 113%; position: relative;" tabindex="0" class="mjx-chtml MathJax_CHTML" id="MathJax-Element-8-Frame">ω in a well-ordered list, it is also true for its container set, the Reals.
"Container set"? I guess you mean "superset". "in a well-ordered list"? a list of what, the naturals?"it is also true...the Reals": by "it" you are saying that an natural number does not "represent" ω, so where are you putting ℝ: are you saying that no natural number "represents" ℝ, or that no real number represents ω? Also, what do you mean by "represent"? You could have a natural number "represent" ℝ as in the Beth-function (Um, how do you get Hebrew letters in this text box?), where Beth-0 is the cardinality of Aleph-0, Beth-n is the cardinality of the power set of Beth-(n-1), and so forth for limit ordinals. So here, the natural number "1" would "represent" ℝ by a peculiar definition of "represent". This is obviously not what you mean to say by the word, but a clear definition would be a good starting point.

If you just want to say that the cardinality of ℝ is bigger than the cardinality of ℕ, that would be easy enough (and you have already represented it in a Venn diagram). But I think you want to say something else, so I will keep needling you to make your terms precise until I figure out what you want to say, and then we can see if you can represent it in a Venn diagram. Unless someone else understands you better. (Think of me as one of those robots in old sci-fi films who needed everything explicitly explained.)
 
  • #8
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Also, what do you mean by "represent"? ... But I think you want to say something else, so I will keep needling you to make your terms precise until I figure out what you want to say, and then we can see if you can represent it in a Venn diagram. Unless someone else understands you better. (Think of me as one of those robots in old sci-fi films who needed everything explicitly explained.)
Thank you for your effort trying to understand me, I'm sorry for my lack of english. I created an image with elementary school logic, hopfully this tells more about what I'd like to explain by a simple Venn diagram (if it is possible to do correctly).

...are you saying that no natural number "represents" ℝ, or that no real number represents ω?
I'd like to say that that no natural number and no real number represents ω, even if the Reals is the superset of the Naturals (and have much bigger cardinality).
 

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  • #9
nomadreid
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I'm sorry for my lack of english.
I didn't notice any lack of English. Your general English is excellent; most native speakers of English aren't used to expressing themselves with the precision (or, to some, pedantry:smile:) required for mathematics.
elementary school logic
I'm impressed. Many countries don't include logic in primary school (alas:frown:).
I'd like to say that that no natural number and no real number represents ω,
Again, we can only get somewhere on this if you can tell me what you mean by "represents".
 
  • #10
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Thank you, but I think at least with difficult topics and mathematical language I have not enough experience, so please feel free to fix my wording if you find obvious errors.
Again, we can only get somewhere on this if you can tell me what you mean by "represents".
I mean some kind of mapping between the sets (just like I tried to demonstrate on the picture). Not exactly bijective, but some kind of a "value keeping" matching, just like between the Naturals and Reals. The connection between Naturals and Reals is not just a simple "one is a subset of the other", but there are more strict rules. By this, if we take the transfinite ordinals from 0 to say omega+1, one by one mapping them to the Naturals, we run out of naturals. If every natural is mapped to a real, and if we keep the rule (that I tried to describe above) then we also would run out of reals.

To make it short, what if my question is:
Is it possible to create a simple Venn diagram that describes the relations between the Transfinite ordinals, the Naturals and Reals? Is there a describable relation at all between them that tries to explain the difference between "up to infinity" and "beyond infinity"? Is this thought have a mathematic sense at all?
 
  • #11
nomadreid
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Let us start with this key fact: ordinal numbers are linearly ordered under set-membership. That is, for any three ordinals α,β,γ, α∈β∈γ⇒ α∈γ. You wish to use subsets, so we can use: for any two ordinals α,β α∈β ⇒ α⊂β. Put the two together, and you get that the subset relation is transitive: α⊂β⊂γ⇒ α⊂γ. So, treating your naturals and reals as ordinals, then there would be a straightforward Venn diagram of all ordinals like the diagram from Wiki that you cited, with the naturals being a subset of the reals being a subset of a higher ordinal, and so forth, and an embedding from one ordinal into a bigger ordinal could be the identity. In other words, the "value-preserving" maps would be fine. (The counter-intuitive thing in the Venn diagram is that two different ordinals, one a subset of the other, can nonetheless have the same size, i.e., cardinality, as one another.) Up to infinity would be the elements of ω, and beyond that would be beyond the smallest infinity. ("Beyond infinity" would not make sense, though.) I think that would answer your question, but if not, try again.
 
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