Everything began from a seemingly simple philosophical question that I asked some times ago:
What are we doing in Set Theory?!
There are several answers for this question. Answers like “we are studying the foundation of mathematics“, but in practice if you look at what appears more than the other topics in Set Theory papers, you find an answer so similar to the following:
The main topic in Set Theory is the study of the behavior of the continuum function under different assumptions including large cardinal axioms.
If we notice that for infinite cardinals is in fact so the above answer becomes as simple as the below statement:
The main purpose of Set Theory is studying the behavior of cardinal exponentiation operator on infinite cardinals.
A “novice” may ask why Set Theory doesn’t study the cardinal addition and multiplication for infinite cardinals? And an “expert” immediately answers:
Because addition and multiplication of infinite cardinals are trivial operations equal to the Maximum function, .
Here there are two counter-intuitive points about addition and multiplication of infinite cardinals that may come to the attention of that “novice” questioner. In fact addition and multiplication operators behave very differently over infinite cardinals with respect to the finite numbers.
- First: The addition and multiplication for infinite cardinals are identical operators (i.e. operator).
- Second: The addition (and multiplication) of two infinite cardinals is equal to at least one of them.
Why? Of course because of their definition which is the “generalization” of what happens in the finite case. The addition of two finite numbers gives a number that is equal to the cardinal of a set formed by disjoint union of two sets with and many members respectively. The multiplication of is also a number that is equal to the cardinal of a set formed by Cartesian product of these two sets. We simply “defined” the addition and multiplication of two infinite cardinals in the same way and then encountered the strange phenomena as stated above.
What about exponentiation? In the finite case is equal to the number all functions from a set with members to a set with members and this is exactly the point that we use for defining the exponentiation for infinite cardinals. But unlike addition and multiplication our defined exponentiation is a highly non-trivial operator in the sense that in most cases for two given cardinal numbers not only alone but also plus large cardinal axioms can’t determine the exact value of what we defined as , the number of functions from a set with members to a set with members.
Here there is a natural question:
What is the source of such a big difference between addition and multiplication of infinite cardinals on one hand and their exponentiation on the other hand? Why the values of the first two operators are trivially determined within but the values of the last are highly undetermined when working in ?
In order to find the answer of the above question we need to analyze the nature of these operators and their connection with each other more carefully. The point is that each arithmetic operator is the iteration of the previous one. Exponentiation is iterated multiplication which itself is iterated addition. For more details see the following question of mine on MSE forum:
The important point here is that there is an infinite sequence of natural arithmetic operators which each of them is defined by iteration of the previous operator. These are called Hyperoperators but are less well known than addition, multiplication and exponentiation because they don’t appear in daily mathematics that frequently.
Now another natural question arises:
Why don’t we define hyperoperators for infinite cardinals? Maybe study of such hyperoperators is as interesting as cardinal exponentiation with possibly much more sophisticated theory which could lead us to discovering several deep independence results.
But before defining hyperoperators for infinite cardinals we need to clarify what we mean by a “valid definition” for such generalized operators. Of course any definition for an arithmetical operator on the infinite cardinals should be consistent with the usual definition of such operators on finite numbers. In this case we can count them as generalization of usual hyperoperators for finite numbers. Also one may want that such a definition satisfies certain “intuitive conditions” like what we have discussed in the above passages.
Here the choice of a “correct definition” in finite case becomes important in the sense that we should choose an intuitively natural combinatorial quantity that our given arithmetic operator represents when we consider it in the finite case. For example we choose the size of disjoint union or Cartesian product as combinatorial quantities that addition and multiplication represent. Then we generalize them to the infinite case to obtain a similar arithmetic operator for infinite cardinals which is agree with the original operator on the finite numbers.
But unfortunately all such intuitive quantities don’t result to the same generalization. So the question for each given hyperoperator reduces to the search for an “appropriate combinatorial quantity” which that particular operator represents. For example exponentiation of two numbers represents the number of functions but it also represents many other combinatorial things, say the number of all trees that have particular properties. Now which one should we choose to be the base of our generalized definition for infinite cardinals exponentiation? Read more in my question and answer here:
For example if we want to define tetration of two infinite cardinal numbers which is the first natural operator after their exponentiation, we first need to find a combinatorial quantity that tetration of two finite numbers is counting it. It is not really as easy as it seems because tetration is a highly rapid operator that barely appears in daily life of a working mathematician and so has few applications. Many just understand it by its definition as iteration of exponentiation. See more discussions in the following question of mine:
Note that direct definitions using recursion don’t work for cardinal arithmetic (but are fine for ordinals) because unlike ordinals, the cardinals are discrete not continuous and you always need to decide what to do at limit stages. If you take supremum in limit steps you never can gain a new operator. For example if we “define” to be which is quite natural if we didn’t know that what exponentiation is in connection with the number of functions, then we get which is against Cantor’s theorem.
In fact in the case of infinite cardinals we never define a higher arithmetic operator based on the defined operators of the lower rank. In the other words we don’t define cardinal multiplication using cardinal addition or cardinal exponentiation using multiplication. We simply give direct definitions for each operator separately, addition is defined by disjoint union and multiplication by Cartesian product and exponentiation using functions.
Now the question is that using what combinatorial quantity should we define the tetration and other hyperoperators for infinite cardinals?
If we find an statement like, “The tetration of two finite numbers is the number of all mathematical objects defined from with such and such properties” then we easily can generalize it to define the tetration operator for infinite cardinal numbers; something so similar to “The tetration of two infinite cardinal numbers is the number of all mathematical objects defined from with such and such properties“.
Then we can compare this newly defined tetration operator for infinite cardinals with exponentiation. Also we possibly can observe that plus large cardinal axioms plus a complete description of the behavior of the exponentiation function over infinite cardinals (say given by ), is still insufficient for determining the value of the tetration operator even in its simplest form, namely tetration of and .
At this moment my main open question in this area is about the intuition behind tetration and the combinatorial quantity that it represents but unfortunately it seems nobody has any idea about what the correct interpretation of tetration of two numbers is in finite combinatorics. I personally contacted several people about this problem but none of them had any idea. It seems the only definition for tetration is given by iteration of exponentiation which based on what is explained here, can’t lead us to a satisfactory generalized definition for infinite cardinals simply because we don’t know how to define tetration in limit stages.
Please let me know if you found some interesting answers for any of the questions stated or linked in this post.