28
votes
How can different models of set theory be constructed from the same set of axioms?
It's helpful to think of the axioms of ZFC as being exactly like the axioms for groups. It just happens that the ZFC axioms are more complicated. Then, different "models" of the ZFC axioms ...
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6
votes
How can different models of set theory be constructed from the same set of axioms?
Then how can there be different models of the same set theory, based on the same set of axioms?
This turns out to be a very general feature of models of first-order theories, due to the compactness ...
- 409k
3
votes
Axioms as PARTIAL information givers of primitive terms - Enderton's Elements of Set theory
The axioms specify properties we are assuming about the undefined terms, but it is not necessarily true that they allow us to answer any question we could ask about the undefined terms. If you start ...
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2
votes
How can different models of set theory be constructed from the same set of axioms?
The theory of mammals includes several axioms:
Female mammals produce milk
Mammals are warm-blooded
Mammals have hair
There are many creatures that satisfy these axioms and are therefore mammals. ...
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1
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Axioms as PARTIAL information givers of primitive terms - Enderton's Elements of Set theory
I suspect you are overthinking (2). Enderton is just applying the absolutely standard definition of logical consequence for sentences in some first-order language $L$ (in this case, the language of ...
- 53.1k
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