This raises the question: ISSUE: Is there some theorem explaining this, or is our vision just more uniform than we realize? Intuition tells me that the power set and replacement axioms hold, as well as choice except in artificial universes , whereas it does not tell me much on the existence of inaccessibles. According to my experience, people sophisticated about mathematics with no knowledge of set theory will accept ZFC when it is presented informally and well , including choice but not large cardinals. You can use collections of families of sets of functions from the complex field to itself, taking nonemptiness of cartesian products for granted and nobody will notice, nor would an -fold iteration of the operation of forming the power set disturb anybody.
So the existence of a large cardinal is a very natural statement and an interesting one and theorems on large cardinals are very interesting as implications, not as theorems whereas proving you can use less than ZFC does not seem to me very interesting. Still, the arguments above are strong enough for me to put them higher than inner models and recognize them for consistency proofs, per se, and also as compared with statements from the AD circle of ideas - this arises for me , in the context of a large ideal on i.
I tend to say yes to this. For me it is an implication, Woodin's view is more or less the inverse. Maybe the following analogy will explain my attitude; we use the standard American ethnic prejudice and status system, as it is generally familiar. So a typical universe of set theory is the parallel of Mr. John Smith, the typical American; my typical universe is quite interesting even pluralistic , it has long intervals where GCH holds, but others in which it is violated badly, many 's such that -Souslin trees exist and many 's for which every -Aronszajn is special, and it may have lots of measurables, with a huge cardinal being a marginal case but certainly no supercompact.
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This seems not less justifiable than stating that Mr. John Smith grew up in upstate New York, got his higher education in California, dropped out from college in his third year, lived in suburbia in the Midwest, is largely of anglo-saxon stock with some Irish or Italian grandfather and a shade of hispanic or black blood, with a wife living separately and 2.
Smith have 2. In this light, L looks like the head of a gay chapter of the Klu Klux Klan - a case worthy of study, but probably not representative. As for the other position - B. For me, the determinacy school is strongly on the syntactical side, being very interested in statements about -sets of reals. Well, I am not so excited by the syntactical flavor of the problems, but more seriously, I agree just that it is a fascinating axiom with a place of honor in the zoo of position B.
L is one, and K, but LA school thinks these answers are wrong, and put them to sleep. Of course, the dispute will not be settled, but it may still be interesting and possible to give the problem some kind of a concrete answer which may be illuminating; I naturally tend to think there may be others.
Note, the fine structure is also syntactical, but it has a lot of consequences which are not, hence ISSUE: how much is the syntactical part needed for applications, e. For Jensen, fine structure is the main point, diamonds and squares are side benefits, probably good mainly for proving to the heretics the value of the theory. Of course, you will need the fine structure for syntactical statements.
Of course not, still there may be positive theorems in this direction. I have strong intuition in favor of both positions, but little knowledge. I also have a keen interest in the natural numbers, though too platonic but not as a set theorist. I will put questions on projective sets under D. For model theory, I will put the zero one laws for sentences in some logic under D.
If you are seriously interested like me in D. With regard to finite support iterations, all regular cardinals bigger than seem to be on an equal footing, but countable support iterations only work for continuum ; when we become interested in this, the preservation of proper forcing [ Sh:b , III] and other properties [ Sh:b , VI] highlight the versatility we have for the case of the continuum equal to. We have many consequences of CH, reasonable ways to prove independence from continuum and a few theorems there is a P-point or a Q-point.
But for the continuum being we are quite in the dark well more exactly, finite support iteration of ccc forcing tell us a lot, but we were spoiled by the better fate of and. Harrington asked me a few years ago: what good does it do you to know all those independences? My answer was: to sort out possible theorems - after throwing away all relations which do not hold you no longer have a heap of questions which clearly are all independent, the trash is thrown away and in what remains you find some grains of gold. This is in general a good justification for independence results; a good place where this had worked is cardinal arithmetic - before Cohen and Easton, who would have looked at?
Now consider cardinal invariants of the continuum, there can be relations between them provable in ZFC which become trivial if the continuum is at most like one being always equal to one of two others ; but the present methods for independence are too weak. If you are interested in D.
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Still, even between people working on boolean algebras and the topology of extremely disconnected compact topological spaces there are differences: are you interested in free sets as an Boolean Algebraist or independent sets as a topologist? The future - the reader may well remind me - what will be the future of set theory? Being optimistic by nature, and proving theorems which look to me reasonably satisfying, I am not at all gloomy.
More seriously, looking at the last hundred years, repeatedly old mysteries have been clarified by deep answers, dark interludes were followed by the opening of new horizons; some directions require a substantial amount of preliminary study while others can be approached with little; and I find the old lady as fascinating as ever. Accordingly I may be foolish, but it is quite hard to prove I am wrong; in any case I have the support of a 20 century trend in history - prejudice is fine, the crime is pretending it does not exist. Also, no originality is claimed - in fact I assume everybody thinks as I do, except when proved wrong.
Secondly, this all applies ipso facto to mathematical logic as I know it. I have little knowledge of recursion theory and considerably less of proof theory, so I refer to model theory and set theory.
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Thirdly, why would anyone want to read this article? You know in your heart that you know better what is important , what is good taste , and so forth. A reasonable guideline may be this: what would I like to be able to read by a reasonable mathematician of Cantor's time? A possible answer: why has he dealt with particular problems even if it was just because his tutor told him to, or perhaps because his tutor told him not to , what were his views - even if not so well-considered as ours, or even self-contradictory, and something about himself and his colleagues.
Note - what a professional philosopher would say should a priori be more coherent , but it is less clear how it is related to what mathematicians do. Locke's books are not necessarily the best explanations to why did Churchill 1 desert James II, nor Rousseau's as to why Robespierre guillotined Danton. So the reader may ask, how do the views here relate to the author's own work? I tend toward A. Hence I think that having a good test problem is usually crucial to the advance of mathematics.
It is to a large extent the duty of new generation to solve the problems of the older one. I thought that while developing classification theory I should try to solve the problems of Keisler and Morley problems which were which made me start my investigations in the first place. This is also the reason for the existence of chapter 14 For Thomas the Doubter in [ Sh:c ]. Even though I thought and still do think that the main gap theorem is the main point, I thought I ought also solve Morley's conjecture as the main gap was my own conjecture, and I did not want to end like the king who first shot the arrow, then drew the circle.
Still, the main gap is called the book's main theorem.
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I suspect that I have the reputation, or notoriety, of emphasizing the value of sport for fun and for competitive value. I do not mean sport for exercise; in fact I find it strange to try and prove known theorems on one's own with casual glances at the existing proof, for the sake of exercise. As I love doing mathematics, I find it more entertaining to solve a problem than to argue about its possible significance, and I have a normal size vanity, so I am also glad to solve a problem just because it is considered hard or important by someone; but even when I know that nobody will be impressed and it may even harm me in some ways, I usually will not refuse the temptation.
Given a choice between working on choiceless universes and sets of reals, with a groan I prefer the latter, and I have dealt with it several times till its solution. I looked at Fuchs' book on abelian groups out of curiosity, as it does not require much background and looks like interesting mathematics, but also because I hoped to find applications of classification theory. As it happened, I found mostly applications of set theory, which strengthened my belief you should usually start from the problems and not the method.
Had my student Mati Rubin not abandoned his assignment to classify the automorphism groups of saturated models of first order theories by interpretability strength to work on the special case of Boolean algebras, I would not have been dragged to [ RuSh 84 ] and then to a long investigation of the quantifier on automorphism of Boolean algebras, etc. Without Cherlin, the non-isomorphic ultrapowers of countable models would not exist [ Sh ] and [ Sh ]. The second quote seems to assert that statements in the object language are essentially commands addressed to the reader by the author.
Neither assertion holds in standard metalogic. Given any valid primary arithmetic expression, insert into one or more locations any number of Latin letters bearing optional numerical subscripts; the result is a primary algebra formula. Letters so employed in mathematics and logic are called variables. A primary algebra variable indicates a location where one can write the primitive value or its complement. Multiple instances of the same variable denote multiple locations of the same primitive value.
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By "logically equivalent" is meant that the two expressions have the same simplification. Logical equivalence is an equivalence relation over the set of primary algebra formulas, governed by the rules R1 and R2.
Let "C" and "D" be formulae each containing at least one instance of the subformula A :. R2 is employed very frequently in primary algebra demonstrations see below , almost always silently. These rules are routinely invoked in logic and most of mathematics, nearly always unconsciously.
The primary algebra consists of equations , i. R1 and R2 enable transforming one equation into another. Hence the primary algebra is an equational formal system, like the many algebraic structures , including Boolean algebra , that are varieties. Equational logic was common before Principia Mathematica e. Conventional mathematical logic consists of tautological formulae, signalled by a prefixed turnstile.
However, conventional logic relies mainly on the rule modus ponens ; thus conventional logic is ponential. The equational-ponential dichotomy distills much of what distinguishes mathematical logic from the rest of mathematics. An initial is a primary algebra equation verifiable by a decision procedure and as such is not an axiom. LoF lays down the initials:. J2 is the familiar distributive law of sentential logic and Boolean algebra. It is thanks to C2 that the primary algebra is a lattice. By virtue of J1a , it is a complemented lattice whose upper bound is.
By J0 , is the corresponding lower bound and identity element. J0 is also an algebraic version of A2 and makes clear the sense in which aliases with the blank page. T13 in LoF generalizes C2 as follows. Any primary algebra or sentential logic formula B can be viewed as an ordered tree with branches. T13 : A subformula A can be copied at will into any depth of B greater than that of A , as long as A and its copy are in the same branch of B.
Also, given multiple instances of A in the same branch of B , all instances but the shallowest are redundant. While a proof of T13 would require induction , the intuition underlying it should be clear.
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