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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 numbers that may include higher order infinities like $\aleph_3$ for example, does that adds any additional structure/properties?

P.S: I'm not a mathematician, but a physicist, so please don't be so abstract or technical.

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I hope that this will answer your question, but I will be happy to add more in here if it doesn't. – Asaf Karagila Oct 28 '12 at 23:44
Thx Asaf but it just explains what is $\aleph$, which I already have an idea about, what I want to understand, is the connection (if any) between them and geometry, the real link between abstract math and physics. – TMS Oct 29 '12 at 20:20

The idea behind cardinality is to mathematically capture the notion of size. What do we want from a notion of size?

  • We want it to be transitive, namely if $|A|\leq |B|$ and $|B|\leq |C|$ then we want $|A|\leq |C|$ as well.
  • We want it to be antisymmetric, namely if $|A|\leq |B|$ and $|B|\leq |A|$ then $|A|=|B|$.
  • We want that if we only changed the "labels" on the elements of a set, then its cardinality will not change. Namely, if $f\colon A\to B$ is a bijection then $|A|=|B|$.

It turns out that we can do that. In modern mathematics the cardinality of a set is the rawest form of size. Think about it as shaking and rattling $\mathbb R$ until we forget its order, the addition and the multiplication. Until we forget the topology and any other structure that we bear in mind when someone says $\mathbb R$. We shake it so hard, nothing is left but a set. This set has a particular size, and its size is $|\mathbb R|$, which we can identify as $\frak c$ or $2^{\aleph_0}$ in most places.

While we can prove that certain properties impose limitations on size (e.g. an ordered field is infinite; a compact and connected metric space has size $2^{\aleph_0}$; etc.) in basic cases, of simply requiring some structure, we cannot really deduce anything.

Furthermore there are two theorems in model theory the upward and downward Löwenheim–Skolem theorems which tell us that under rather loose conditions we can ensure that a first-order theory has models of any cardinality.

For example, given any infinite set -- regardless to its size -- we can prove there is an operation which makes it into a group; or a ring; or even a vector space over $\mathbb R$ over any dimension which obeys the limitations imposed by cardinality. If we can do that on any infinite set, given an arbitrary cardinal $\aleph_\alpha$ we can take a particular ordinal which we often identify with $\aleph_\alpha$ (the $\alpha$-th initial ordinal) and simply endow that ordinal, that set, with such structure.

If we wish to discuss naturally arising structures then we need to assume the existence of some structure. However assuming the existence of some structure is not part of what we envision from cardinality. Cardinality just tells us "how many cats are in the bag", and nothing about their state.

However we may wish to still identify the natural structure an ordinal has. As we define ordinals to be the set of those ordinals smaller than itself (so $\alpha<\beta\iff\alpha\in\beta\iff\alpha\subsetneq\beta$) this internal structure is mostly just order, but it could also be ordinal addition and multiplication.

For example $\omega_1$ (the first uncountable ordinal, identified with $\aleph_1$) has the property that if $\alpha,\beta<\omega_1$ then $\alpha+\beta,\alpha\cdot\beta,\alpha^\beta<\omega_1$ (where the last one is ordinal exponentiation). So in this sense $\omega_1$ has more than just order, it has natural operations which we can require to be continuous with respect to the topology. We can ask what is the coarsest or finest topology which has such property (that ordinal arithmetics is continuous with respect to it). We can also discuss about continuous functions from $\omega_1$ (or any other ordinal, really) to $\mathbb R$ or any other topological space.

While ordinal spaces have properties which are directly linked to the ordinal itself (e.g. $\omega_1$ as a space is a sequentially-compact, non-compact, first-countable space, solely for the reason it is $\omega_1$). However this does not answer, in my opinion, on your question. This structure does not directly relates to us "using higher cardinalities", but rather to us using an ordinal space of uncountable cofinality without a maximal point.

It does, however, answer the question "What sort of other interesting topological structures one can naturally find in mathematics except topologies on finitely dimensional vector spaces over $\mathbb R$?" and that is a whole other question.

One nitpick, though, it is quite possible that $\aleph_3$ is much much smaller than the size of the real numbers. In fact we cannot prove too much on the cardinality of the continuum, except that it is big. How big? Well, we don't know, but quite big.

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Thx Asaf for the detailed answer, anyway it made me somehow confused: we can ask what is the fines topology that can satisfy that, but what is the answer? I mean that if want now to reformulate the question to become what is the cardinals that may impose kind of finest topology that has natural, continuous order with compactness support, (or like you formulated it) should I make it as new question? Also you confused me with $\aleph_3$ being smaller than real numbers size, isn't Contour proof that depends on one-to-one correspondence is showing that? – TMS Oct 31 '12 at 20:06
@TMS: First the easy part, Cantor's proof shows that the real numbers have cardinality strictly larger than $\aleph_0$. However it was proved that we cannot compute the exact cardinality in the usual axioms of set theory (read: ZFC). For example the real numbers could have the size of $\aleph_{42}$, or even higher. As for the rest of the comment, I just don't understand what exactly you are trying to say. – Asaf Karagila Oct 31 '12 at 20:09
@TMS: As for the confusion about the size of the continuum, you might want to start by reading this:… – Asaf Karagila Oct 31 '12 at 20:11
Yes I understood you now. I asked also if I should post new question "What sort of other interesting topological structures one..." you wrote or maybe you will answer it here? – TMS Oct 31 '12 at 20:14
@TMS: I see. Well, for example $\omega_2$, as an ordinal space, is not just sequentially compact, but every sequence of $\aleph_1$ points has a converging subsequence which contain $\aleph_1$ points. It is still not compact, despite having quite strong properties which "limit" infinite sets (much like how in a compact space every infinite set has a limit point). The problem is that your question is not well-phrased and it is hard to understand. Maybe if you think about it and figure out exactly what you are trying to understand, then ask again [...] – Asaf Karagila Oct 31 '12 at 22:15

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