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What I'm trying to find:

A concrete example in predicate logic with the following characteristics:

  1. An infinite system of axioms for a finitely axiomatizable theory.
  2. Any finite subset of the infinite axiom system should no longer axiomatize the theory.

My first intuition was to recursivly define a infinite set of formulas which axiomatize the theory, but I didn't manage to build any examples myself. So I'm not sure if such a axiomatic system is even possible

Alex Kruckman
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how_is_this_supposed_to_run
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  • Please include some more context in your question, such as where you got it or what you have tried. – Greg Nisbet Jun 29 '21 at 15:28
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    Can'1 you just add infinitely many tautologous sentences? – Bram28 Jun 29 '21 at 15:28
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    Add $\forall x (x = x)$, $\forall xy (x =x \land y = y$, $\forall xyz(x = x \land y = y \land z = z)$, etc? – eyeballfrog Jun 29 '21 at 15:29
  • Apologies for the badly written Question, English is not my first Language and I'm having trouble correctly phrasing what I'm after.

    yes @Bram28 that would be a viable solution to how I phrased the Question.

    What I really want is:

    1. An infinite System of Axioms for a finitely axiomatizable theorem.
    2. Any finite subset of the infinite axiom system should no longer axiomatize the theorem.

    Sorry again, and thank you for taking your time to answer

     – how_is_this_supposed_to_run Jun 29 '21 at 17:03

1 Answers1

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This is impossible, by the compactness theorem. See my answer here: https://math.stackexchange.com/a/3814809/7062

Alex Kruckman
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  • I didn't close the question as a duplicate because the question asked is different, even if the answer is the same. If others (or a moderator) disagree, feel free to mark this as a duplicate! – Alex Kruckman Jun 29 '21 at 17:18
  • Thank you for your answer! A quick follow up question: this means there is no axiomatic system for a basic elementary class which meets the characteristics mentioned in my original question? – how_is_this_supposed_to_run Jun 29 '21 at 17:33
  • Generally the criterion for duplicates is based on answers, not questions. (The message when a question is marked as a duplicate is not "This question was asked here:" but "This question has an answer here".) If it's not completely obvious how the linked duplicate answers a question, you can briefly explain that in a comment. (And if the explanation is too long for a comment, then probably you want to write an answer instead of just marking it as a duplicate.) – Eric Wofsey Jun 29 '21 at 17:34
  • That said, the specific question you picked here is probably not the best choice of a duplicate for this question. I'm sure something very close to this exact question has been asked several times before. – Eric Wofsey Jun 29 '21 at 17:35
  • @EricWofsey Well, if you can find a better duplicate - as I said, feel free to mark it. – Alex Kruckman Jun 29 '21 at 17:39
  • Yeah, I'm looking for one now. It actually seems surprisingly hard to find...maybe I'm just not searching for the right phrases. – Eric Wofsey Jun 29 '21 at 17:39
  • @EricWofsey I extensively tried to search for a answer to my question before posting, but couldn't find any. Maybe it was my language barrier, or I didn't know how to phrase it with the right terminology. – how_is_this_supposed_to_run Jun 29 '21 at 17:40
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    @how_is_this_supposed_to_run I don't see how your follow up question differs from your original question. In case it's not clear: There is no theory $T$ satisfying conditions 1 and 2 in your question. – Alex Kruckman Jun 29 '21 at 17:42
  • @AlexKruckman okay that clears it up, thanks! – how_is_this_supposed_to_run Jun 29 '21 at 17:43
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    Hmmm, it seems I may have just been wrong. Lots of questions with answers that use this idea have been asked but I actually can't find one that is really on the nose. – Eric Wofsey Jun 29 '21 at 17:48