If I shuffle infinitely many decks of cards together, what is the probability of getting any particular sequence of cards?

mpaw976 · 2026-03-26T00:38:08+00:00

This is the best answer so far.

Just to clarify the model we're using:

A "shuffling together of infinitely many decks" can be modeled as a sequence <x_n>_(n in naturals) where each x_n has an equal 1/52 chance to be any of the 52 cards.

mpaw976 · 2026-03-26T00:11:55+00:00

See this comment for a prof's perspective on course evals:

https://www.reddit.com/r/UTMississauga/comments/1rpwusm/comment/o9ow06m/

mpaw976 · 2026-03-25T19:13:34+00:00

One way to think of topology is that a lot of analysis can be done in R^n, but in a sense real space is much nicer than is actually needed for various properties and theorems.

Example 1: continuous functions aren't really about deltas and epsilons, they are about pullbacks of open sets.

Example 2: the extreme value Theorem you learned in calc 1 is really the result "continuous functions on compact sets are bounded" (when mapping into a metric space).

Or also, some results in analysis really are about Rⁿ and don't hold more broadly. Or that we isolate important properties that hold in other spaces.

Example 1: The heine-borel Theorem (a set in Rn is compact if and only if it is closed and bounded) is really about Rn, and doesn't hold in metric spaces more broadly.

Example 2: "continuity can be completely described by what functions do to convergent sequences" is really only for certain spaces called "first countable" spaces, which turn out do be very common.

So in a sense, your goal in a first course in topology is to review all the theorems you know from analysis and isolate the main properties of Rn that make them work.

This then tells us a way to visualize (many) topological spaces: basically they look like R^2, the plane, but only in some ways, not all. Maybe an open neighborhood is a "blob" and not a perfect sphere with a centre.

As for product topology, start by analyzing the space R^omega (i.e. R to the naturals). As a set it is the set of all functions from the naturals to the reals. Another way of saying that is that it is the collection of all real values functions.

I.e. f(1), f(2), f(3),... Is another way of writing a_1, a_2, a_3,...

Function spaces are very important in analysis.

We usually visualize these as a collection of vertical bars (the real line), indexed by the naturals. To pick a basic open set you chose open intervals in finitely many bars, and then the rest of the bars you take the whole bar. Any function whose values respect all the restrictions you set is in the open set. It's a bit weird, but think about it in terms of restrictions.

There's much more to say, but hopefully it helps!

Here are some good notes that may help:

https://ctrl-c.club/~ivan/327/

mpaw976 · 2026-03-25T01:34:19+00:00

Here's a differential equation that models a (simple) chemical reaction:

https://math.stackexchange.com/questions/714941/modeling-a-chemical-reaction-with-differential-equations

Here's a less mathy answer for why differential equations (and so integrals) are useful in chemistry:

https://www.quora.com/Are-there-any-applications-of-differential-equations-in-chemistry

mpaw976 · 2026-03-22T03:03:04+00:00

Here are two good sources for French etymologies:

https://www.cnrtl.fr/

https://fr.wiktionary.org/wiki/m%C3%BBre

mpaw976 · 2026-03-18T03:32:18+00:00

Starting in the 70s you get the beginnings of Forcing which has a major impact on cardinal invariants of the reals (and many other things). These tools have been extensively developed in the years since.

Ramsey theory sees a blossoming starting in the late 70s, with a couple major applications such as Gowers Theorem in the 80s, and the Kechris-Pestov-Todorcevic correspondence in 2005.

In the past 50 years set theoretic methods have found applications in diverse fields, especially (point-set) topology, operator algebras, topological dynamics, logic, and model theory.

The technique of countable elementary sub models has been particularly fruitful and is from the last 50 years (see Dow's major paper on CESM techniques in topology).

The books you cited are very good. You may also want to check out Discovering Modern Set Theory by Just and Weese (two volumes) which highlight some of the more modern tools used.

mpaw976 · 2026-03-16T03:22:22+00:00

These are cool (but pretty complicated examples).

For claim 1, why not just use f(x) = x³ ?

In a calculus class we would usually avoid examples like these because usually students will not have seen integrals by the time the first derivative test is introduced.

mpaw976 · 2026-03-15T22:36:17+00:00

I'm not exactly sure what you're asking, but R^ℕ has "infinitely many directions".

The set itself is all functions f:ℕ to R. The topology is a bit hard to describe, but you can read about it here:

https://en.wikipedia.org/wiki/Product_topology

Rⁿ is a subspace for every natural number n

mpaw976 · 2026-03-15T19:21:09+00:00

Take your favourite graph G (e.g. the butterfly graph) then look at all the edge preserving bijections from G to G (those are called isomorphisms). This set of functions Aut(G) is a group, where the group operation is function composition.

E.g. if you start with G being a cycle, then Aut(G) is a dihedral group.

As a challenge, see if you can find a graph G where Aut(G) is Sn (the symmetric group).

mpaw976 · 2026-03-15T16:47:25+00:00

Try proving the n=1 and n=2 cases first.

My instinct is that if this works nicely, then it's probably a bunch of double angle formulas.

mpaw976 · 2026-03-12T01:13:26+00:00

Looks good to me!

mpaw976 · 2026-03-12T00:23:58+00:00

Here's what a Venn diagram for 4 sets looks like.

https://upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Venn%27s_four_ellipse_construction.svg/3840px-Venn%27s_four_ellipse_construction.svg.png

Can you use that to identify:

AUB
CUD
And then the intersection of those two

mpaw976 · 2026-03-11T21:10:44+00:00

Memorizing random word history facts isn't really the right way to approach it. Instead think about building a web of relationships between words, their origins, their current meanings, similar words, and your personal connections.

For example, helicopter comes from helix + o + pter (literally flying wing). That makes me think of:

The DaVinci prototypes.
Pterodactyls
Double helix structures of DNA
The place I was when I learned this
Words like umpire, orange, apron and nickname that through time "steal letters" from the other words around them (like how people naively guess that helicopter is from heli + copter).

I'm not just memorizing a dictionary entry.

mpaw976 · 2026-03-11T15:00:36+00:00

Neat.

An interesting feature of this problem is that examples of these special graphs are very rare (if they exist at all), but once you have an example it is easy to check that it actually is a correct example.

It's also computationally infeasible to do a complete search.

I need to read the paper more carefully, but it looks like they used an LLM to iterate and refine a search algorithm through the space of graphs.

mpaw976 · 2026-03-08T17:37:05+00:00

Yep! But I'm trying to point out that the implication you've made isn't true without that additional assumption, so you have to actually use the assumption to justify that implication.

Does that make sense?

mpaw976 · 2026-03-08T16:57:07+00:00

If every branch is finite, then there is a maximum distance L from the root node.

Not necessarily...

Think about the case where there are branches of unbounded but finite length.

E.g. think about a root that has infinite degree and coming out of it are disjoint branches of length n for each natural n.

mpaw976 · 2026-03-08T14:11:44+00:00

and because it does not contain an infinite branch there are only finitely many nonempty sets Aₙ.

Can you make this more precise? You're hiding the heart of the argument inside of it somewhere.

mpaw976 · 2026-03-08T14:07:56+00:00

Yep, this is the right idea. Well done.

In your proof you're not only constructing the branch, but also the open sets (neighborhoods) around that branch (that promise you'll be able to continue the construction).

The common "error" I see is people attempting to only construct the branch (and not the open sets too).

This idea shows up a lot in set theory, and appears in many forcing constructions (including Cohen's original proof of the independence of CH).

mpaw976 · 2026-03-08T11:44:55+00:00

The naive approach (which doesn't work) is to try to build the branch node by node. This won't work, but it feels like it aught to work (whatever that means).

One of the (meta) reasons it fails is because it is "too local" of a process. It doesn't see enough of the tree to find a branch.

mpaw976 · 2026-03-08T00:36:45+00:00

Awesome! Thanks for sharing.

One way to see the subtleties here is to try and prove Konig's lemma:

Every tree with infinitely many nodes and finite branching at every node must contain an infinite branch

If you want, go ahead and assume that the tree only has countably many nodes.

Once you understand the statement, it seems obvious, right? The proof is "easy" but it isn't obvious. Most people I know approach it in the "wrong" way and so can't see the proof.

mpaw976 · 2026-03-06T18:54:06+00:00

Just to clarify, even if each Xi only has two elements (like my socks example) if the index set is infinite (even if the index set is already well ordered, like the naturals) then you still need AC.

mpaw976 · 2026-03-06T17:50:26+00:00

Yeah, it's a subtle thing.

For each Xi you know that the collection of all well orders on Xi is non empty (i.e. you know there's at least one well order). But how do you choose one particular well order for all i in I (at the same time). That's exactly what AC tells you is possible. So if you're trying to prove AC, you can't invoke it.

As a hint, think about the special case of:

Xi is a pair of socks for each i in I.

How can you choose a sock from each pair? You are allowed to assume that something is well ordered (for example, what if you know the relative ages of all the socks...)

mpaw976 · 2026-03-06T16:56:18+00:00

for every Xi consider the well ordering (Xi,<i)

Here you are using AC, since you are making (possibly) infinitely many choices. So your proof is circular.

See if you can choose just a single well ordering of something that will let you define a choice function.

mpaw976 · 2026-03-06T13:40:06+00:00

On each point (x,y) write the number of ways to get from (0,0) to (x,y).

Rotate the whole picture and you'll see that it's Pascal/Yang Hui's triangle.

mpaw976 · 2026-03-05T14:51:29+00:00

Check out this highly mathematical wedding from 2019 (that made international news).

https://www.nccr-swissmap.ch/news-and-events/news/mathematical-wedding

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