Robinson V. Davis, 2: Fill & Download for Free

The answer to this question varies wildly by statement and by person.Let’s look at some examples.Con(ZFC)Before we say anything else, let’s remind ourselves that something can be independent of ZFC only if ZFC is, in fact, consistent. An inconsistent theory proves everything, so nothing can be independent of it. In other words, when people say “CH is independent of ZFC” or “Large cardinals are independent of ZFC” they always quietly mutter “assuming ZFC is consistent, of course. Duh.”But of course, Con(ZFC) is neither provable nor refutable in ZFC if ZFC is consistent, by Gödel’s Second Incompleteness Theorem, so it is independent. Strangely, you can say that it’s only independent if it’s true! That’s just the way it is.I wouldn’t say anyone thinks Con(ZFC) is “neither true nor false”, and in fact I would say that virtually everyone would say that it’s true. The fact that ZFC can’t prove it is not seen as a serious argument for it not having a truth value.Famously, Vladimir Voevodsky put forth the suggestion[1] that ZFC might be inconsistent, but he openly admitted that this was more of a way to instigate a discussion about foundations. Tragically, he is no longer with us to explore the foundations of mathematics and his vision for them, including Homotopy Type Theory. I do hope others will continue to venture down that road.Among working mathematicians outside of set theory, the notion that ZFC might be inconsistent would seem outlandish, and if it’s the case it’s because of some obscure interaction that is far removed from day-to-day math, one which can probably be mended by slightly restricting the axiom of comprehension or something.To summarize (here and below, the numbers are for illustrative purposes only – I have very little data to back any of this up.)True: 99.9%, False: 0.1%, Indefinite: 0%.V=LThis is the axiom of constructibility[2], studied by Gödel as part of his exploration into the consistency of AC and GCH with ZF. Indeed, V=L implies both AC and GCH, and it further settles many other independent assertions.Does V=L? Most people who believe this is an answerable question would say No (as Shelah wrote[3], “Why the hell should it be true?”). I’m frankly not aware of any set theorist who seriously argues that the answer is a plain and simple Yes, though of course it’s possible that some people believe just that and they don’t consider it important to notify me.In contrast with Con(ZFC), however, it’s a lot less clear that there is a matter-of-fact with respect to this question. Proponents of a “multiverse” view of mathematics, such as Joel David Hamkins and (I believe, to some extent) Saharon Shelah, would simply say that it’s equally legitimate to explore set theories with and without V=L (and any other axiom). Gödel would have disagreed.In summary:True: 0%, False: 60%, Indefinite: 40%.CH and GCHI’m lumping the Continuum Hypothesis and the Generalized Continuum Hypothesis together. Of course it’s possible for someone to hold different opinions with respect to either of these assertions’ truth value and definiteness, but they largely go together.Does the Continuum Hypothesis carry a truth value? Is there a fact-of-the-matter with respect to whether there is or isn’t a family of sets of natural numbers which is uncountable yet not in bijection with the reals? There is by no means a consensus on this.Solomon Feferman published (just a few years ago!) a paper[4] titled “The Continuum Hypothesis is neither a definite mathematical problem nor a definite logical problem”. Peter Koellner disagrees[5]. Raymond Smullyan said “…some very formalistic mathematicians say this means that the Continuum Hypothesis is neither true nor false. Well, I don’t buy that and neither would most – many – mathematicians. We want to know whether the Continuum Hypothesis is true or not.”In brief, opinions vary. Among those who believe the question has an answer, I think most tend towards “False”, but certainly some people believe that CH is true. David Gale reports on a brief survey in the last chapter of “Tracking the Automatic Ant”. According to him, most non-set-theorists saw it similarly to the parallel postulate: true in some models, false in others. Some set theorists leaned more platonically, though even among those there was no clear consensus on whether it’s true or false.I think it’s something like:True: 5%, False: 25%, Unsure: 20%, Indefinite: 50%.Martin’s Axiom[6]I’m putting this one here mostly to say that I have no idea what people think about the truth value of MA([math]\kappa[/math]). It’s an axiom which lives in the vacuum created by CH: if CH is true, MA is true as well. It only becomes interesting if CH is false, and then the question becomes at which cardinal precisely (between [math]\aleph_0[/math] and [math]\mathfrak{c}[/math]) Martin’s condition ceases to hold.Just how many sets with Lebesgue measure [math]0[/math] do you need in order to force a positive measure? Continuum-many surely suffice, countably-many surely don’t. What happens in between, assuming that there is an “in-between”? I wonder if someone who believes [math]\lnot[/math]CH is agnostic about this one. I don’t know. I expect many people think this has a definite answer, but I don’t know.True: ?, False: ?, Indefinite: ?Large Cardinal AxiomsThe large cardinal axioms[7] present a very different “kind” of ZFC-independence, one that Shelah says make the universe of sets “taller” rather than “fatter” (the latter is the realm of Forcing[8], as in the independence proofs of AC and CH).As is well-known, large cardinal axioms tend to arrange themselves linearly along a degree of consistency strength. Therefore, if you believe any of them is “true”, you must also believe that all smaller large cardinals are also true, and vice versa for “false”. Therefore, if anything, more people are likely to believe in the smaller large cardinals than in the large ones.My personal impression is that if a mathematical proof genuinely necessitates an inaccessible cardinal, most mathematicians would accept that proof, but would feel a strong desire to remove this dependency. For a while, some people even considered Wiles’ proof of FLT to be in this state (due to its reliance on “Grothendieck Universes” at some point), but I don’t believe this is still a concern. It is now accepted the the proof can be formalized in ZFC, and likely even PA.I think many people are agnostic with respect to the reality of large cardinals. They find it interesting to explore their relationship and consequences without worrying to much about whether a superhuge cardinal does or doesn’t “exist”. People who aren’t quite agnostic would be less agnostic about anything below measurable cardinals, and may more agnostic about anything above.I don’t think any mathematician considers any large cardinal to outright not exist. But again, there’s no reason for anyone to keep this random dude on the internet posted on their philosophical positions, so I may well be wrong on this.In summary, depending on the size of the cardinal:True: 1%-40%, False: 0%, Indefinite: 60%-99%Diophantine EquationsAfter delving deep into the unknowable, we should go back and remind ourselves that ZFC can sometimes be surprisingly powerless to prove some very mundane assertions. Con(ZFC) was one example, but here’s another (essentially equivalent one): it follows from the work of Davis, Robinson, Putnam and Matiyasevich that there exist Diophantine equations with no solutions which ZFC can’t prove are solution-free.Those are nothing more nor less than concrete polynomial equations with integer coefficients for which we seek solutions in integers. Yes, there are such equations for which ZFC can’t decide if they do or don’t have solutions.The truth value of such assertions is hardly in question. Those equations can have a solution in integers iff ZFC is inconsistent, which very few people seriously believe is an option. So, they don’t have a solution, but we just can’t prove that they don’t in ZFC. Just something we need to accept. Nobody ever promised that our recursive theories are going to be complete. They’re not.So, the assertion that such a polynomial equation does not have a solution in integers would be regarded as:True: 99.9%, False: 0.1%, Indefinite: 0%.Yup. Wild variation.Footnotes[1] What if Current Foundations of Mathematics are Inconsistent?[2] Axiom of constructibility - Wikipedia[3] Logical Dreams[4] https://math.stanford.edu/~feferman/papers/CH_is_Indefinite.pdf[5] http://logic.harvard.edu/EFI_Feferman_comments.pdf[6] Martin's axiom - Wikipedia[7] Large cardinal - Wikipedia[8] Forcing (mathematics) - Wikipedia

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Resources that are pretty good are Yahoo.com, ESPN.com. FFToday, CBSportsline, Pro Football Focus, The Huddle and Rotoworld.Lots of sites do player rankings - it would be great if a site did a blended average, like RCP does for political polls, but I am unaware of that existing.Other tips:+ If you do not get Rodgers, Brees, Brady, Cam or Stafford in round one, wait on a QB. Rivers, Peyton and Eli, Rivers, Romo, Ryan, Vick, Schaub, Cutler .. you can get these players in mid rounds. Take a gamble on greatness LATE with RG3, Luck or Russell Wilson, as all play on teams that will be trailing a lot. Guys like Dalton, Bradford and Fitzpatrick will be available late, too, and are decent #2 quarterbacks and situational plays (whenever they face the Rams, lol).+ The only TEs worth an early grab are Gronk and Graham. Vernon Davis still has Alex Smith at QB; Witten just smashed his spleen; Gates has struggled to stay healthy; Pettigrew, Gresham, Hernandez, Celek, Finley, Rudolph, Davis, Winslow.. You can get production in middle rounds. Two sleepers: Tamme in Denver, and Keller (NYJ). Tamme caught 67 passes in 10 games the last time he played with Manning in Indy. Keller is playing for a new deal, and in the last year of his contract, and other than Holmes, the Jets have few appealing targets for Sanchez.+ Do not overdraft Adrian Peterson. He is 10 months off an ACL injury - usually that is an 18 month deal. Plus, he is 27. Guys that come back off ACLs at or near 100 percent tend to be younger - Jamal Lewis did it at 23, and ran for 2,066 yards two years later. But guys who are older who blow a tire - Jamal Anderson for example - usually drop off 15-20 percent. Meaning best case, A.P. is a below-average fantasy #1 back, but a good #2. Also: he plays on turf, behind a poor line, with a young QB (Ponder). None of that is good for him in 2012. It hurts me to write this, because I love A.P. as a player, but I will hurt YOU more if you take him before round three - at the earliest.+ Be wary of WRs in year one with new teams. V. Jax should be great in TB, B. Marshall should be very good in Chicago, where he reunites with Cutler, and Robert Meaham figures to exel in SD, and Brandon Lloyd should be really good in NE. Pierre Garçon in DC, Laurent Robinson in Jville, Mario Manningham in SF, Eddie Royal in SD.. I cannot endorse any of them as sure things. Moss in SF and T.O. In Seattle? That is anyone's guess.+ I really like WR Torrey Smith, A.J. Green, Julio Jones, Victor Cruz, Daris Heyward-Bey, Demaryius Thomas, Greg Jennings and Percy Harvin for 2012. My hunch is they all will outperform their draft positions. Calvin Johnson and Larr Fitzgerald will be great. Hakeem Nicks is coming off a foot injury; Andre Johnson and Wes Welker, though great players, at on the wrong side of 30. DeSean Jackson, Santonio Holmes, Dwayne Bowe.. All could boom or bust. I know I will overdraft Stevie Johnson, because I love the Bills, but that is my guilty pleasure this year. Sue me.+ I'm not keen on Michael Vick because he will not have all-universe LT Jason Peters blocking for him this year. True, Vick is a LH QB, but he will miss Peters -trust me on that. His replacement, Bell, is a HUGE downgrade.+ RBs are a mess. LeSean McCoy, Arian Foster, Matt Forte, Ray Rice - no problem. MJD coming off a holdout is now an injury risk - that hammy or groin pull is never late for RB or WR off long layoffs. Mike Wallace at WR - same thing, be careful. CJ2K and Reggie Bush will be good, but until Jake Locker and Ryan Tanneyhill show they can sling it, these backs will see eight man fronts every week. Run DMC could be a bust, or the #1 back in fantasy - I will not be shocked either way. Ryan Mathews just got hurt; Frank Gore, Willis McGahee, Steven Jackson and Michael Tuner have a lot of miles on their tires, and when it goes for RBs it goes vy fat, so be careful; Fred Jackson is terrific, but figures to split work with also terrific C.J. Spiller in Buffalo; Stevan Ridley, DeMarco Murray, Marshawn Lynch, Beanie Wells, Jamaal Charles, Isaac Redman, Shonn Greene.. You tell me.+ I know we just saw Cam Newton do amazing things, but rookie QBs rarely light the league up. Rookie WRs rarely do much. Rookie RBs do, however, so do not sleep on Martin, Wilson, Richardson (pending that knee injury), etc.+ Remember most of all you are NOT drafting who players HAVE BEEN but rather, drafting what they WILL BE. RBs and WRs over 30 are not a good idea. Try to get explosive young players, ages 23-28, on teams with good QBs and OLs.Good luck, and good drafting.