Let S and T be mathematical sets. Now consider the following two propositions:
1. S is a proper subset of T.
2. S and T have the same number of elements.
Are (1) and (2) consistent? That is, can they both be true? If yes, explain how.
If you think (1) and (2) are consistent, then consider whether there is anything to the following analogy. If there is, explain the analogy. There is a set G. G has three disjoint proper subsets, F, S, H. All four sets agree in cardinality: they have the same number of elements.
If S and T are infinite, (1) and (2) are consistent. O/w they are not.
Example:
Let S be the set of all even numbers; let T be the set of all positive integers. S is a proper subset of T, since 1 is a member of T, but not of S. However, S and T have the same cardinality.
The same cannot hold for any finite sets S and T.
Right. But what about the analogy?
If God is infinite, then certain subsets of God are also infinite (and also God), those being the Father, the Son, and the Holy Spirit?
The idea is that God is to the Persons as an infinite set is to three infinite disjoint proper subsets. They have the same cardinality: aleph-nought. That models the divinity common to God, Father, Son, and Holy Spirit. The disjointness of the proper subsets models the distinctness of the Persons. That there is one superset models the oneness of God.
That’s the analogy.
But is it a good one? Does it render the Trinity doctrine intelligible and contradiction-free?
It depends on what we mean by each person of the trinity is God. If the ‘is’ is the is of absolute identity, then it won’t work, for the three disjoints proper subsets are not absolutely identical to the superset. So the question is what is the analog of “being the same cardinality as the superset”? Are we inching here towards relative identity?
Peter,
That none of the subsets is absolutely identical to the superset seems to show that the analogy fails.
Can you think of a good clear example in which a and b are the same F but not the same G? And don’t say that the Father and the Son are the same God but not the same Person! For if that were the only clear example of sortal-relative identiy, then we would not have a good explanation of the coherence of the Trinity doctrine.
Bill,
I have tried the example of the mass substance water in another tread. We can say that there is the same amount of water in three smaller buckets as in one big one with identical volume as the sum of the three smaller ones. So we can count water in terms of its volume (the same quantity of water) or in terms of the number of container in which the quantity of water fits (the number of buckets). I have not there worked out the full analogy I had in mind.