/sci/ - Science & Math

hurr infynitee is not exzist
hurr biolojee=hard maths

yes, but only theoretically. its cant be used in real live.

I was under the impression that aleph-one is a cardinal number, not an ordinal number, and that ordinal numbers are used as index numbers

>>4244441
confirmed for full retard

you dont know what your talking about.
im taking ecology right now and fish population dynamics, and, beleive me its hard.
also your ovbiously a troll because, math is easy compared to real science.

>>4244417
no, that notation doesn't make any sense. The cardinals are index by the ordinals, but aleph sub omega_1 does exist.

>>4244441
Cardinal numbers are just certain special ordinal numbers.
For example, Aleph0 = omega

If we accept that <span class="math">\aleph_0[/spoiler] exists, and if we accept that <span class="math">\aleph_n[/spoiler] exists for all <span class="math">n\in\mathbb{N}[/spoiler], then it's easy to see that <span class="math">\aleph_{\aleph_0}[/spoiler] exists, right? But there really isn't an extension beyond this, it seems...

Can someone explain how <span class="math">\aleph_{\aleph_n}[/spoiler] exists for any natural n greater than zero?

>>4244417
yes, <span class="math">\aleph_{ \omega }[/spoiler] does exist and is import as it's the first uncountable cardinal such that the cardinality of the reals is 100% not equal to in ZFC.

>>4244478
which ordinal number corresponds with beth-one ?

>>4244479
>But there really isn't an extension beyond this, it seems...

Yes there is.
Alephs are defined such that Aleph(a+1) is the smallest infinite ordinal not equinumerous with Aleph(a), with Aleph(0)=omega, and Aleph(L) for some limit ordinal L is the smallest ordinal not equinumerous with any Aleph(M), for M<L.

So your cardinal is perfectly well defined.

>>4244487
ZFC doesn't give an answer.

>>4244532
It seems like beth-one should not be an ordinal number. The ordinal numbers are transitive sets, but beth-one contains {6}. 6 contains 5, but {6} does not contain 5, which violates transitivity. So I don't think beth-one is an ordinal number.

>>4244417
Define "exist". It exists in some kinds of math, and not in others.

>>4244553
No, it is an ordinal number, just not a specific one.
Beth-one could be aleph-one or aleph-two or three or four..., but not aleph-omega, as >>4244486 said

>>4244580
then where is the flaw in my argument?

fuck your jewish physics

>>4244588
>which violates transitivity

It doesn't, because 5 is contained elsewhere in bath-one

>>4244553
Note that I'm not saying that Beth-1 IS an ordinal, just that it has an ordinal associated with it.
Like how the rationals are equal to the natural numbers, but are associated with the same ordinal.

>>4244417
>thinly veiled meta-metaphysics troll
3/10 OP.

cool pic though. props.

>>4244553
beth_1 = 2^(aleph_0) = |P(aleph_0)| != P(aleph_0)