Timeline for Arguments against large cardinals
Current License: CC BY-SA 2.5
5 events
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| Oct 29, 2010 at 23:25 | comment | added | arsmath | Nothing in the OP commits you to Platonism. Axiomatic set theory functions more at the level of meta-mathematics than mathematics -- we want to be able to say a mathematical notion is well-defined by defining it in set theory. For example, the p-adics are well-defined because we can define them in set theory. I don't have an opinion on whether the p-adics "really exist" in some Platonic sense. If the idea of measurable cardinals doesn't imply a contradiction, then they seem like a well-defined notion, so we'd like to be able to exhibit an example, but within ZFC we can't. | |
| Oct 29, 2010 at 21:52 | comment | added | Timothy Chow | I don't wish to contest your philosophical position; however, I believe that your questions about children and computer programmers are misguided. I could just as well argue that since most mathematicians don't know what PA is (they have some vague concept of the Peano axioms, but most would stumble over a precise statement of what we mean by a first-order induction scheme), then PA can't be fundamental. The question of what makes for satisfactory logical foundations is totally different from the question of what we should teach practitioners, whether they be children or mathematicians. | |
| Oct 29, 2010 at 20:58 | comment | added | Lucas K. | That is exactly the problem of Hilbert's program. I am doing research if it is possible to extend PA with some axiom scheme (not using cardinals), such that the relative consistency between this extended PA and ZFC can be proven. Again, the question is not whether the set of natural numbers exists, but in which way they must be part of the fundamentals. And, if ZFC + large cardinals is fundamental, why don't we learn our children these axioms on primary school? And how could it be, that millions of computer programmers, capable of some mathematical reasoning, don't know these axioms? | |
| Oct 29, 2010 at 20:33 | comment | added | Stefan Geschke | In the philosophy that you describe there are no large cardinals. But is there the set of natural numbers? An infinite set does not exist in a very concrete sense, either. However, the nonexistence of the set of natural numbers would make a large part of mainstream mathematics difficult. So where should we stop accepting strong axioms? To say we do accept ZFC (which implies the consistency of PA and is therefore strictly stronger than PA), but nothing beyond ZFC, seems to be artificial. | |
| Oct 29, 2010 at 20:04 | history | answered | Lucas K. | CC BY-SA 2.5 |