Synthetic argument in mathematics
Have a good week Rabbi,
I recently heard about the continuum hypothesis in mathematics and heard that it has been proven that it cannot be proven false and that it cannot be proven true. Does this mean that there may be synthetic claims in mathematics? That is, claims about which there is no and cannot be certainty, but which have some truth value?
Best regards,
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0 Answers
Yes. This is a famous example of Gödel’s theorem regarding incompleteness (weakness, I think) in mathematics. But there is no place for synthetics in mathematics. What is synthetic is not mathematical. Once it has been proven that both possibilities are consistent, it ceases to be a hypothesis and becomes a free choice. The choice itself is not a mathematical act. Gödel showed that there is also strong incompleteness (which is a theorem that is true and cannot be proven), and this is closer to what you are looking for. But the name of the theorem is also true because it has been proven, except that the proof is done outside the axiomatic system in question.
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