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Consider the infinite set of propositions {0+0=0, 0+1=1, 0+2=2, 0+3=3, 0+4=4,...}. It seems clear that one knows all of them, because, they all follow from "0 plus any natural number is that natural number". But, how can this be, when it seems that everybody has a finite number of beliefs?

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  • Because we know the general proposition ∀n (0+n=n) of which the above are particular instances. Commented Mar 5, 2018 at 15:49
  • See Peano axioms. Commented Mar 5, 2018 at 15:49
  • We can know schemas of propositions such as "n = n". But since our brains are finite, and our lives are finite, we can only know finitely many individual facts or propositions or thoughts.
    – user4894
    Commented Mar 6, 2018 at 3:12

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One standard way to deal with this sort of situation is to distinguish between explicit and implicit beliefs (See Gilbert Harman's Change in View, Ch.2). Say that a belief is explicit if one possesses a mental representation whose content is that belief, and say that a belief is implicit if it is not explicit, but is (easily) inferable from one's explicit beliefs. Thus, we can say that you implicitly believe each member of the given set as long as you explicitly believe, say, the Peano axioms.

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  • Agreed. I would say the natural numbers aren't finite, but they are "finitely describable"-- they can be described with a finite number of terms.
    – user935
    Commented Mar 5, 2018 at 19:43
  • @barrycarter. "...with a finite number of terms," assuming that at least one of those terms represents an infinite concept. Notice Mauro's expression: ∀n (0+n=n). It has a finite number of terms, but the universal operator quantifies over an infinite domain.
    – user3017
    Commented Mar 6, 2018 at 13:19

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