Naive Set versus Axiomatic Set Theories

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The power set will be a major component of our connection between sets
and the laws of thought. While we will carefully avoid getting lost in
too much algebra, we'll find it convenient to give them their own symbol
and algebra if only to simplify the text itself. We will therefore call
the power set and refer to the power set of a set as .
We will also need to think about the power set of a power set and so on:

As it is our plan to consider thought only in the context of the *real Universe* we need a very concrete set to play with to figure out
what is and how it works. Consider, therefore, the set consisting
of four cards pulled out of an ordinary deck of playing cards. To make
differentiating easy, we'll pull out the four aces and consider each
card to be labelled by its *suit*.

Our toy set is thus:

(3.1) |

Here is a listing of formed from the permutations of the four
symbols taken 0 to 4 at a time (where order doesn't matter and each
object can only occur once in a set):

(3.2) |

where the first entry is the

There are two general kinds of things we can ``do'' algebraically with in terms of thought, reason, and language. One is that we can identify particular sets from by means of a suitable predicate expression. For example, I can ``create a set that has one card that is a black suit and is not a spade'' to uniquely define the set .

There will often be many ways to create a predicate that specifies a
single subset from the power set, but there is one way that *always*
will exist. We can always specify the subset by explicitly specifying
the list of its members^{3.13}. We will call this method ``identification'' as it appears to be
somehow related to the law of identity. Note that we use identification
of the elements of the original set , plus the processes of
permutation and union to generate . It seems difficult to
imagine - literally - working with a set whose members cannot be
identified independent of predicates used to describe them.

The second kind of thing we can do is to *identify* (in precisely
this sense or via predicates) particular *sets* of subsets drawn
from the . If we specify (for example) ``the set of all sets in
that contain a heart'' we end up with:

(3.3) |

Note that there is no way to collapse or reduce this to a member of the original power set. Each of these sets is an object in its own right that satisfies the criterion for selection.

This set of subsets (drawn from ) is itself a subset from a set
of subsets of the subsets of the original set ^{3.14}. Clearly this set is . There seems to be no reason we
cannot similarly recursively generate for any finite by
iterating the process of making out of the sets
generating by permuting the members of taken from 0 to the
cardinality of times^{3.15} .

Of course, this process scales fairly agressively. We cannot actually
*draw* even because the number of elements in it is
, and the number of elements in the
is
and so on. However, if the cardinality of
the original set is finite, so is the cardinality of the
for any finite . It's just large.

This may seem like a rather lot of complexity - in only the third level
power set we already have considerably more objects than atoms in the
physical Universe, for example, and it was *only* four cards!
However, nothing less will do, as the answer to *any set theoretic
question we can ask* must lie therein. Fortunately for us all, in the
physical Universe a great deal of this complexity can be compressed by
the human mind into *structure*.

We have already performed such a simplification - imagine if we specified in terms of the very large set of molecules that make up the cards, of the even larger set of atoms that make up the molecules, of the larger still set (call it, say, ) of of elementary particles (electrons and quarks and the various field quanta) that make up the atoms and their nuclei. The first level power set would contain many absurd (non-physical) subsets, but it would also include subsets that contained just three quarks and an electron, which on a good day could take on a new name: a ``hydrogen atom''. Indeed, follow the process of forming power sets forward, we will discover therein sets of sets of elementary particles that aggregate into other atoms, sets of sets of sets that aggregate into molecules, and so on up to cards.

So each of our cards is *actually* internally organized into
structures that can be treated as independently identifiable subsets,
themselves aggregated into independently identifiable subsets, all part
of a whole hierarchy of . The card is just one out of a
*very large number* of such subsets, with all sorts of internal
symmetries. The count of permutations, and permutations of
permutations, etc. scales up extremely rapidly, which is why statistical
mechanics works as well as it does in physics. There is no infinity
there, but there are plenty of finities that (as I like to tell my
students) are really *good friends* with infinity, their children
play together, every now and then they all get together at infinity's
house and drink a few beers.

We are therefore fortunate indeed that the human brain more or less
automatically makes this sort of hierarchical decomposition when
confronted with permutative power set-theoretic information that even at
the first or second levels causes our internal number-registers to beep
and return ``overflow''. And this is still, recall, *just four
cards*. Imagine dealing with a *deck* of cards, or a Universe with
*many* decks of cards that are one tiny part of one tiny planet in
one small solar system in a single galaxy. Yet when I refer to each of
these things, your mind *effortlessly* erases all the detail and
replaces it with a *hierarchy* drawn from power set upon power set
all the way down to whatever the *real, existential* microscopic
elementary set of objects are that make up the Universe (where we might
have to include all of the points in space and time some way in our set
descriptions.

There are a number of consequences of this hierarchical decomposition.
One to keep in mind is that when we reason about anything *real* (as
opposed to mathematics, which might be real, might not - lots of
controversy there and I don't want to get into it) we are forced to do
so at the level of one of these , and maybe a few recursions
on either side of it. We *cannot* extend our reasoning down to the
indefinitely microscopic or up to the indefinitely macroscopic. It is
absurd to try to understand the rules of poker in terms of the
properties and motion of the elementary particles of the Universe *even though every particle in the game* obeys rules defined at that
level at *all times*. Nor do we compute the effect of folding a
hand in the poker game on the motion of the Milky Way galaxy as it
meanders around in the gravitational field of all the other galaxies in
the Universe. More *is* different, and so is *less*.

For this and many other excellent reasons that we'll go into, our *actual* reasoning process about the *actual* Universe is almost
immediately forced to be *probabilistic*. This suggests that when
we get around to axioms and all that, one of the *first* things we
should work out is the mathematics of *induction* as the process of
building the hierarchies is necessarily inductive as *otherwise
there is no reason to favor any particular decomposition over any
other*. We must *find* a reason, or give up on ``reason''
altogether.

For the moment, though, let's *ignore* all this appalling complexity
and go *back* to just the some given finite set and maybe
and , just to see what insight we can gain from this
formulation into the laws of thought.

Set Theory and the Laws of Thought

Naive Set versus Axiomatic Set Theories

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