Trinity, logically described

In this post I want to make clear my position on so called “trinity” of God. I am a mathematician and won’t write nonsense like “the Son is identical to the Father, but they are logically distinct”.

What I will formulate is nearly logically rigid, but not quite as I don’t specify in which logical framework I do the “accounting”. I assume that my accounting is logically correct, but I am not 100% sure that what I will formulate is logically consistent and “good” for describing God (isn’t appearing to compare God with lower things than Christ). I propose to check my theory, by formulating it in mathematical rigid and (dis)proving its logical consistency.

I do believe in Trinity. However I prefer another formulations (without pejorative using number “three” to describe God) like “Christ is full content of God”. See my book New Testament Commentary by a Mathematician for biblical (and thus without using here the number three) description of how Christ relates to God and what is Christ in his essence. In this post I however will describe it in regard of trinity, despite I do not like this word to describe God.

I will call two objects A and B “predicate-equivalent” if and only if there is a bijection f mapping all predicates P of one variable true for the argument A into all predicates Q of one variable true for the argument B, such that P(x) is true if and only if (f(P))(x) is true for every variable x and predicate P of one variable.

Hm, still unsure whether EVERY two objects are predicate-equivalent, so making this setting unsuitable for description of trinity of God. But I will put forth my preliminary thoughts in the hope we will reach more exact knowledge then.

I claim that Father, Son, and Holy Spirit are pairwise predicate equivalent.

Note that the equivalence in the previous paragraph implies that there are other objects equivalent to Father, Son, and Holy Spirit. For example such objects are the set {Father, Son, and Holy Spirit} and the triples (Father, Son, Holy Spirit) and (Holy Spirit, Son, Father). I expect that however if we limit the set of objects to compare by predicate equivalence to “living persons” (whatever this may mean mathematically) then this set should have exactly three elements: {Father, Son, and Holy Spirit}.

In mathematics it seems to make sense to identify equivalent (predicate-equivalent in this case) objects to say they are one the same object. For example positive whole numbers can be identified with natural numbers. In the same sense we can (in some but not in all logical frameworks) identify Father and Son. The question whether the Son is identical to the Father is thus dependent on the used logical framework.

I have formulated some properties of Trinity nearly with mathematical rigid. Let us now study it from positions of mathematical logic. At first we need to make sure that not every two objects are predicate-equivalent.

My thoughts on this are very preliminary. I thought I know it well, but when attempted to formulate it to write in this blog post, I found that I do not yet understand this thing.

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