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Within an infinity of possibilities there is a possibility that God exists in all possibilities just as much as there is a possibility that God exists in none of them. This leads to a contradiction. Does it mean that an infinity of possibilities is not possible?

The potential possibilities are included. This is not about God's (non)existence. The concept of God is used to illustrate the thought experiment. The focus is on the question and not on the (non)existence of God.

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    Perhaps it means that there is more than one meaning of possibility: the possibility of the potential but unrealised, and the possibility of the realised but unknown, and that they may not be equivalent. Which do you think the existence of gods falls under? – Niel de Beaudrap Feb 14 '15 at 15:11
  • Thanks for sharing Niel. I don t think that is relevant because the unrealised posibilities and the unknown realised one fall within the total. I rather not go into god's (non)existence. God is used as an example to illustrate the thought experiment. – Dan Coman Feb 14 '15 at 16:46
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That god is a possibility means that there is at least one possible world with a god.

That god is not a possibility means that there are no possible world with a god.

That it is not impossible that there is no god means that there is at least one possible world with no god.

That it is impossible that there is no god means all possible worlds have a god

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You are using the term possibility in two distinct ways, one to refer to possible worlds and the other to refer to all possible worlds.

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This is not an answer, but I can't add a comment yet so here are my thoughts.

First, why is this a paradox? It seems your question begins to fall on probability. If so, there's no paradox there, that is simple the game of life. In theory you can flip a coin an infinite amount of times and get either heads or tails on each flip. The frequency with which you will get either a heads or a tail eventually will fall on some limiting value, as the number of flips increases.

This may only begin to become a paradox, given one's absolute conviction on the existence of a God, because the fact that there is a God and the fact that there isn't a God are equally probable events in all cases. But usually, you don't have someone with absolute conviction in some form of God's existence who will also be convinced about the 50/50 chance of God existing.

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