Larry, interpret the smiley face as saying:
PA + (◻C → C) |-
Then the steps 1-10 of Lob’s proof would seem to take us from:
PA + (◻C → C) |- (◻C → C)
to
PA + (◻C → C) |- C
If this were the case, the deduction theorem would in fact come into play and give us:
PA |- (◻C → C) → C
So there must be at least one step of the reasoning 1-10 that does not apply. Which one and why?
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Larry, interpret the smiley face as saying:
PA + (◻C → C) |-
Then the steps 1-10 of Lob’s proof would seem to take us from:
PA + (◻C → C) |- (◻C → C)
to
PA + (◻C → C) |- C
If this were the case, the deduction theorem would in fact come into play and give us:
PA |- (◻C → C) → C
So there must be at least one step of the reasoning 1-10 that does not apply. Which one and why?