Timeline for The sets in mathematical logic
Current License: CC BY-SA 3.0
20 events
| when toggle format | what | by | license | comment | |
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| Nov 30, 2016 at 9:35 | comment | added | მამუკა ჯიბლაძე | My excuse for the belated comment: the question has been bumped up :D Now what I want to say - it seems to me you are actually using some kind of induction principle here, by believing that our finitary manipulations with finite strings will never lead to something bad, since we know that if they worked so far then nothing will fail if we add some extra finite amount of data. And induction principles are not something I would call self-evident. | |
| Sep 27, 2012 at 5:56 | comment | added | Stefan Geschke | @Todd: Yes, I get that. And I agree. I guess what I really wanted to say is this: Computer Science shows the success of the method. At Goedels time one might have argued that there was circularity in the foundations of math and that this was somehow problematic, but now the "physical workings outside our mental models" witness that everything is working just fine. | |
| Sep 26, 2012 at 16:55 | comment | added | Lee Mosher | @Michael Greineker: Speaking as one who plays a dwarf in a popular massive multiplayer on-line role playing game, touche. My real point, though, was that contrary to @Mirco (apologies for earlier mis-spelling), I myself am not able to play these mathematical games without having some form of deeper belief in the reality of what they represent. It is the "blindness" of the game, which seems accepted in this thread, to which I am objecting. | |
| Sep 26, 2012 at 16:35 | comment | added | Todd Trimble | Hi Stefan; by luck I just happened to see your comment. I am inclined to regard that possible circularity as somewhat spurious: the mathematical models might give a mathematically literate programmer, who wants to design a proof checker, a useful conceptual guide for what she is doing, but the physical workings of the machine are outside whatever mental models she is using. (But I guess you completely get that and I'm essentially repeating myself.) | |
| Sep 26, 2012 at 16:09 | comment | added | Qfwfq | (btw, in line with neo-positivism I personally don't believe mataphysics makes any sense, but some people do) | |
| Sep 26, 2012 at 16:06 | comment | added | Qfwfq | As it was remarked above, we must start somewhere, and we (mathematicians) choose to start from believing that we can conceive symbols (say "$0$" and " $1$") and trust manipulations with them, without any "logic" or set theory involved, just blind copying and substituting strings according to some rules. What is behind this blind game (whether it's a human mind, a Platonic heaven or a computer) is not a matter of mathematics but of phylosophy of mind, metaphysics, or physics. | |
| Sep 26, 2012 at 14:42 | comment | added | Stefan Geschke | @Todd Trimble: I looked at this question again after a while, since there were some new comments. I think you have a very valid point. I personally think that today a lot of truly formal mathematical logic is much accessible than at Goedel's time, because everybody is more or less familiar with how computers work. On the one hand, the machine can check proofs, on the other hand, we have good mathematical models how the computer works. This somehow indicates the possible circularity, but it also shows that there are probably no problems with that. | |
| Sep 26, 2012 at 14:20 | comment | added | Michael Greinecker | @LeeMosher: One can write a huge novel about hobbits and orcs without believing in their existence in any deep sense. | |
| Sep 26, 2012 at 13:05 | comment | added | Lee Mosher | @Marco: Perhaps one can "play" ZFC without any such belief. Or perhaps one can play topology or differential geometry or PDEs. But I kind of doubt it. I think that if I did not let my mind open up to some belief in the reality of the objects modelled in the states of its neurons, I would not get very far in proving anything about those objects. | |
| Jun 14, 2011 at 18:56 | comment | added | Mirco A. Mannucci | Stephan, your answer is GOLD. Yes, one must start somewhere, and what better place than finite strings and their manipulations? I, for one, do not believe in either sets or (platonic) numbers, but I have no problems with manipulating strings (my laptop does not have any issue either, and "it" does not know a thing of numbers and sets, unless by numbers ones means terms and their verifiable equivalence under the rules of arithmetics and term rewriting). The funny (and great) thing is: one can "play" ZFC without believing that anything there has any substance beyond syntactic games... | |
| Apr 25, 2011 at 9:00 | vote | accept | zzzhhh | ||
| Apr 24, 2011 at 23:40 | comment | added | Todd Trimble | +1 Stefan: "what we are using is some intuitive understanding of how to manipulate finite sequences of characters from a finite alphabet." I find it helpful to consider a computer which is programmed to detect whether a formal proof in a formal theory is valid. The proof is expressed in finitely many symbols, and there is no background set theory sitting inside the computer on which the proof-checking depends. The pattern of flow of electrons through the logic gates is an extra-linguistic entity, and this breaks the circularity. | |
| Apr 24, 2011 at 21:18 | comment | added | Stefan Geschke | Actually, I wouldn't say we are using any axioms of set theory. What we are using is some intuitive understanding of how to manipulate finite sequences of characters from a finite alphabet. If you want to formalise this, some fragment of number theory will be enough (as finite sequences can be coded as natural numbers). | |
| Apr 24, 2011 at 18:59 | comment | added | zzzhhh | So the axioms of set theory used in mathematical logic is: only finite set is set (can be an element), while other infinite set (we should call them class) is not a set (can not be an element), even if it is allowed to be a set in ZFC, right? | |
| Apr 24, 2011 at 18:04 | comment | added | Stefan Geschke | A (proper) class cannot be an element of a set or class. It can, however, be a subclass of another class. Induction over proper classes is no problem. In number theory (where the only objects, i.e., elements of the realm of discourse, are numbers) all numbers form a proper class, yet you can do induction over this class. In ZFC the ordinals, a proper class, form a subclass of the whole universe. Induction over the class of ordinals is no problem, neither is induction over any other class with a well-founded, set-like relation (like $\in$) on it. | |
| Apr 24, 2011 at 17:19 | comment | added | zzzhhh | Do you mean by "class" an object that can not be an element of some class or set? If so, can a class be a subset or subclass of another set or class? I think no, which would, however, makes induction by structure of wffs impossible. For example, in propositional logic, the class of all propositional wffs (it's infinite) is defined as a smallest subset of words satisfying .... | |
| Apr 24, 2011 at 16:32 | comment | added | Stefan Geschke | You can define ZFC and provability in ZFC without actually assuming the existence of infinite sets. The collection of sentences that makes up the theory ZFC is then a class, not a set. This is a somewhat extrem point of view, but I don't see any problems with it. If you want to prove more advanced theorems of first order logic like the completeness theorem you will have to assume more. Concerning your question, my strong opinion is 1) There is no paradox, in the sense of contradiction, just circularity as commented above. (Roughly the same as "PA is consistent".) | |
| Apr 24, 2011 at 14:07 | comment | added | zzzhhh | Finite set is OK, but there are also many infinite sets used freely in mathematical logic, how to transfer from finite set to infinite set safely without fearing any paradox? And, may I ask what is your exact opinion: 1)No paradox (why?), 2)There is paradoxes, but we have successfully excluded them (how?), 3)Not clear, but only hold a belief that there is or there is not any paradoxes? | |
| Apr 24, 2011 at 13:18 | history | edited | Stefan Geschke | CC BY-SA 3.0 |
added 575 characters in body; added 98 characters in body
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| Apr 24, 2011 at 13:04 | history | answered | Stefan Geschke | CC BY-SA 3.0 |