Bernard Bolzano was a Bohemian (i.e. from Bohemia, not Gypsy [this remark is for the sake of clarity only, for the meaning of Bohemian can be easily confounded by an English-speaking reader]) mathematician, philosopher and theologist. He addressed the problem of existence of infinite sets. First he has shown that there indeed exists an infinite ammount of something. Then from the existence of God he has shown the
actual existence of a infinite set.
The proof goes as follows: (beware: matemathical induction ahead)
Theorem 1. There exists at least one valid theorem.
proof: by contradiction. Let there be no valid theorems. Then the theorem
``There is no valid theorem'' is valid. Q.E.D.
Theorem 2. There exists infinitely many valid theorems.
Proof: follows from Lemma 3.
Definition 1. Let Tk denote a theorem ``There are at least k valid theorems,'' for integral k > 1, furthermore let T1 denote ,,There is at least one valid theorem''
Lemma 3. For every integral k > 0 the theorem Tk is valid.
Proof: by induction. As it has been shown, there is at least one valid theorem, namely ``There is at least one valid theorem.''
Therefore T1 is valid.
Let n be integer and Tk be valid for all integral k > 0, n > k.
We do ask, whether Tn is valid. Suppose it is not. Then there is a valid theorem stating ``There are no more than n-1 valid theorems,'' which is absurd [face=SPIonic]
o(/per a)/topon[/face].
Now some useful mathematical properties of the God. (postulated as axioms [note: this axioms are postulated for this proof only, your axiom set and/or belief concerning entity and/or entities commonly referred to by similar names may be different]
The God will be referred to as ``the God'', and ``G'' hereinafter.
Axiom 1. There is the God.
Axiom 2. The God knows everything.
Axiom 3. The God can think of everything.
Theorem 4. There
actually exists an infinite set.
Proof: The God can think of all the Tk at once (from A3). He does so (A2)
((as G knows all Theorems that are true, G knows all Tk as well, therefore they are in G's mind, therefore G thinks of them))
Therefore there actually exists a set of Tk in G's mind, if G exists and has the properties A2 and A3.
Therefore (A1) there exist an infinite set, for we have shown that the number of Tk is not finite.
More information on Bolzano can be found eg. here:
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Bolzano.html.