I wonder if there are natural examples of sets whose cardinality is $\frak p$
and $\frak t$?
Examples that could provide intuition for non- set-theory experts?
I find the definitions in the Malliaris & Shelah paper that established
$\frak p=t$ difficult to grok, so to speak.
I am not asking to understand the proof, just to understand the cardinals themselves.



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> Malliaris, Maryanthe, and Saharon Shelah. "Cofinality spectrum theorems in model theory, set theory, and general topology." *Journal of the American Mathematical Society* 29.1 (2016): 237-297. 
>  ([Journal link](http://www.ams.org/journals/jams/2016-29-01/S0894-0347-2015-00830-X/home.html); [Link to earlier arXiv version](https://arxiv.org/abs/1208.5424).)
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Related MO question:
[Short proof of $\frak p=t$](https://mathoverflow.net/q/168125/6094).



Intuition for the infinite cardinals p and t (now that p = t)?
Each element of **p** and each element of **t** is a countable family of infinite sets of natural numbers.
Let's start by giving an example of such family.


A simple example of a countable family of infinite sets of natural numbers would be $$E = \bigotimes\_{j\in \Bbb N} \{k \in \Bbb N : \exists m | k = jm \}$$ so that two elements of $E$ would be $\{ 2,4,6,8\cdots\}$ and $\{ 3,6,9,12\cdots\}$. We can say that a countable family of infinite sets of natural numbers is a subset of $[\Bbb N]^{\aleph\_0}$. (We will see that this particular $E$ is not in **p**.)


Now let's define a relationship between sets $A$ and $B$ which I will call "is almost a subset of," written as $A \subseteq^\* B$. 
In words, $A$ is almost a subset of $B$ iff only a finite set of elements of $A$ are not also in $B$. That is, 
$$A \subseteq^\* B \Leftrightarrow |\{ x:x\in A \wedge x \not\in B\}| < \infty $$ For example if $A$ is the set of primes and $B$ is the set of odd natural numbers, then $\{ x:x\in A \wedge x \not\in B\} = \{2\}$ and $A \subseteq^\* B$. 


Note that it is quite possible for $A \subseteq^\* B$ and $B \subseteq^\* A$.




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Now let's define **p** and give an example of one element of **p**.


**p** is the set of all countable families of infinite sets of natural numbers such that family $f \in \mathbf{p}$ if and only if two conditions are met:


* Every non-empty finite subfamily of $f$ (every finite collection of sets, all of which are in $f$) has an infinite intersection.
* But there is no infinite set of integers $A$ such that $A$ is almost a subset of each element of $f$. That is, $\left( \forall A \subseteq \Bbb N: B \in f \implies A \subseteq^\* B \right) \implies |A| = \infty$.


You can see that these two conditions work against one another in some sense. A simple example of a family of sets $f\_1$ meeting the first condition is 
$\left\{ \{ n+0 :n \in \Bbb N\},\{ n+1 :n \in \Bbb N\}, \cdots \right\}$ since the intersection of any finite number of elements of $f\_1$ consists of all numbers $\geq$ the highest starting point of any of that finite set of elements.


But that family $f\_1$ fails the second condition, because if $A$ is the (infinite) set of even integers, then $A$ is almost a subset of each member of $f\_1$. 


An example of an element of **p** is the family of sets (indexed by $k\in \Bbb N$) of the form $\{ m^k : k\in \Bbb N\}$. The first condition is met since if $\ell$ is the least common multiple of all the indices in the finite sub-family 
then the set $\{ m^\ell : m\in \Bbb N\}$ is in every set in the finite subfamily. The second condition is met since there are an infinite number of integers that are cubes but not squares and vice-versa, so any infinite set $A$ of integers is either not almost a subset of the squares, or not almost a subset of the cubes, unless all but a finite number of elements of $A$ are sixth powers. But then $A$ won't be almost a subset of the fifth powers, unless all but a finite number of elements of $A$ are $30$-th powers; and so forth.




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Now let's define **t** and give an example of one element of **t**.


**t** is the set of all countable families of infinite sets of natural numbers that can be well-ordered by the $\subseteq^\*$ relation. (Remember, well-ordering will mean that if family $T$ is in **t** then every (non-equivalent under $A \subseteq^\* B$) pair of elements of $T$ can be related by $\subseteq^\*$ in one direction or the other, and every subset of $T$ has a least element in that ordering.)


A simple example of an element of **t** is 
the family $f\_1$ described above. This "tower" happens not to be in **p**.


The **p** = **t** question, then, is whether **p** can be mapped onto **t**, or **t** can be mapped onto **p**, or they are in $1:1$ correspondence.




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In the paper by Malliaris and $(\mbox{Shelah})^2$ there is a minor wording error in definition 1.1 whch might be causing some confusion: The first bullet should read 



> 
> $D$ has an **infinite** pseudo-intersection if there is an infinite $A\subseteq \Bbb N$ such that $\forall B\in D, A \subseteq^\* B$. 
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