Three's

SQL_lall

Quote:

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aleph-0 + 1 = aleph-0
also: aleph-0 x 2 = aleph-0 (I think)
That's what happens with infinity

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Actually, reading up a little on transfinite numbers, that is not the case.

(aleph-0) + 1 = (aleph-0) + 1
AND
1 + (aleph-0) = (aleph-0)

Also

(aleph-0) x 2 = (aleph-0) x 2
AND
2 x (aleph-0) = (aleph-0)

What you are actually stating above is that (aleph-0) x 2 has the same cardinality as (aleph-0).

i.e. ====== =========
(aleph-0) =(aleph-0) x 2

Simonm: Are you are asserting that add & multiply are not commutative when applied to an integer and a transfinite number?

Quote:

---

Actually, reading up a little on transfinite numbers, that is not the case.

(aleph-0) + 1 = (aleph-0) + 1
AND
1 + (aleph-0) = (aleph-0)

Also
(aleph-0) x 2 = (aleph-0) x 2
AND
2 x (aleph-0) = (aleph-0)

What you are actually stating above is that (aleph-0) x 2 has the same cardinality as (aleph-0).

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As far as I know, (Aleph-0) * 2 has the same cardinality as (Aleph-0). Why do you have the pairs of statements suggesting lack of commutativity?

BTW: The following site has an easy to read, but lengthy, essay about transfinite numbers. It addresses the above, and seems to state that adding a finite number or multiplying by a finite number does not change the cardinality of a transfinite set.

http://www.earlham.edu/~peters/writing/infapp.htm

In that essay, no distinction is made between

1 + (Aleph-0) and (Aleph-0) + 1

or between

2 * (Aleph-0) and (Aleph-0) * 2

Do you have some reason to think there is a distinction?

Guv

There is some confusion here over ordinals and cardinals.

Quote:

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1 + (Aleph-0) and (Aleph-0) + 1

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The cardinality of the two expressions above is equal and I did not mean to convey a contrary opinion.

I was saying that 1 + (Aleph-0) is the same ordinal as (Aleph-0). Whereas, (Aleph-0) + 1 is a different ordinal. The point being that the order of terms in the expression does matter when dealing with transfinite calculations.

I will admit that I am entirey basing that on what I have read in "Infinity and the mind" (Rudy Rucker) and I am pretty sure that is what was conveyed in that book. I may be wrong (it wouldn't be the first time) and if I am, I would readilly like to know in what way I have misinterpreted it (or, indeed, been mislead).

I haven't yet viewed that link that you provided but I will when I get more time.

Simonm: Interesting coincidence. I too, have a copy of Infinity and the Mind by Rudy Rucker, and had no problem finding the pertinent discussion.

You are correct: There is confusion on my part about ordinals and cardinals.

From what he says, your statements are correct about ordinals, but not about cardinals.

It never occurred to me that there would be such a difference between transfinite ordinals and transfinite cardinals. It is another of those weirdities that I will never really grok about this subject matter.

I just plugged 'analemma' into google and found this interesting site:
http://www.analemma.com

It seems that this is a word to describe a shape like a figure eight, but not the name of the infinity or the figure eight symbols.

oops! missed a page, sorry.