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In a previous question, I asked whether there can be effectively axiomatizable set theories (at least as strong as, say, ZF) that are complete modulo first-order arithmetic, to which the answer is no; there will be sentences of second-order arithmetic that can't be proved even using the omega-rule. I'm now curious whether there can be set theories that are complete modulo finite-order arithmetic (in the comments of my last question, someone claimed the answer was no without proof, but he has since deleted his comment). That is, for any effectively axiomatizable set theory at least as strong as ZF, will there be a sentence independent of it with no arithmetic consequences? For ZF specifically, the continuum hypothesis would be an example of such a sentence, but that can be easily rectified by adding it as an axiom.

In a previous question, I asked whether there can be effectively axiomatizable set theories (at least as strong as, say, ZF) that are complete modulo first-order arithmetic, to which the answer is no; there will be sentences of second-order arithmetic that can't be proved even using the omega-rule. I'm now curious whether there can be set theories that are complete modulo finite-order arithmetic (in the comments of my last question, someone claimed the answer was no without proof, but he has since deleted his comment). That is, for any effectively axiomatizable set theory at least as strong as ZF, will there be a sentence independent of it with no arithmetic consequences? For ZF specifically, the continuum hypothesis would be an example of such a sentence, but that can be easily rectified by adding it as an axiom.

In a previous question, I asked whether there can be effectively axiomatizable set theories (at least as strong as, say, ZF) that are complete modulo first-order arithmetic, to which the answer is no; there will be sentences of second-order arithmetic that can't be proved even using the omega-rule. I'm now curious whether there can be set theories that are complete modulo finite-order arithmetic (in the comments of my last question, someone claimed the answer was no without proof, but he has since deleted his comment). That is, for any effectively axiomatizable set theory at least as strong as ZF, will there be a sentence independent of it with no arithmetic consequences?

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Set theories that are complete modulo finite-order arithmetic

In a previous question, I asked whether there can be effectively axiomatizable set theories (at least as strong as, say, ZF) that are complete modulo first-order arithmetic, to which the answer is no; there will be sentences of second-order arithmetic that can't be proved even using the omega-rule. I'm now curious whether there can be set theories that are complete modulo finite-order arithmetic (in the comments of my last question, someone claimed the answer was no without proof, but he has since deleted his comment). That is, for any effectively axiomatizable set theory at least as strong as ZF, will there be a sentence independent of it with no arithmetic consequences? For ZF specifically, the continuum hypothesis would be an example of such a sentence, but that can be easily rectified by adding it as an axiom.