He worked with the Prime Number Theorem and Riemann's Hypothesis;and proved the unexpected fact that Chebyshev's bias, and, while true for most, and all butvery large, numbers, are violated infinitely often.
One of his prize citations commends his "sheer technical power,his other-worldly ingenuity for hitting upon new ideas,and a startlingly natural point of view."Much of Tao's work has been done in collaboration: for examplewith Van Vu he proved the circular law of random matrices; with Ben Greenhe proved the Dirac-Motzkin conjecture and solved the "orchard-planting problem."Especially famous is the Green-Tao Theorem that there are arbitrarily longarithmetic series among the prime numbers (or indeed among any sufficientlydense subset of the primes).
LOGICAL PROOF OF CONTINUUM HYPOTHESIS - …
His "exotic" 7-dimensional hyperspheresgave the first examples of homeomorphic manifolds that were not alsodiffeomorphic, and developed the fields ofdifferential topology and .
in a proof that the Hypothesis is false.
Let the generic multiverse conception of truth be theview that a statement is true simpliciter iff it is true in alluniverses of the generic multiverse. We will call such a statementa generic multiverse truth. A statement is said tobe indeterminate according to the generic multiverseconception iff it is neither true nor false according to thegeneric multiverse conception. For example, granting our largecardinal assumptions, such a view deems PM (and PD andAD(ℝ)) true but deems CH indeterminate.
Continuum hypothesis - Wikipedia
This motivates the shift to views that narrow the class ofuniverses in the multiverse by employing a strong logic. For example,one can restrict to universes that are ω-models, β-models(i.e., wellfounded), etc. On the view where one takes ω-models,the statement Con(ZFC) is classified as true (though this is sensitiveto the background theory) but the statement PM (all projective setsare Lebesgue measurable) is classified as indeterminate.
Can the Continuum Hypothesis Be Solved? | Institute …
There is a form of radical pluralism which advocates pluralismconcerning all domains of mathematics. On this view any consistenttheory is a legitimate candidate and the corresponding models of suchtheories are legitimate candidates for the domain ofmathematics. Let us call this the broadest multiverseview. There is a difficulty in articulating this view, which may bebrought out as follows: To begin with, one must pick a backgroundtheory in which to discuss the various models and this leads to adifficult. For example, according to the broad multiverse conception,since PA cannot prove Con(PA) (by the second incompleteness theorem,assuming that PA is consistent) there are models of PA + ¬Con(PA)and these models are legitimate candidates, that is, they areuniverses within the broad multiverse. Now to arrive at thisconclusion one must (in the background theory) be in a position toprove Con(PA) (since this assumption is required to apply the secondincompleteness theorem in this particular case). Thus, from theperspective of the background theory used to argue that the abovemodels are legitimate candidates, the models in question satisfy afalseΣ-sentence,namely, ¬Con(PA). In short, there is a lack of harmony betweenwhat is held at the meta-level and what is held at theobject-level.
05/01/2018 · Can the Continuum Hypothesis Be Solved
The pluralist is generally a non-pluralist about certain domains ofmathematics. For example, a strict finitist might be a non-pluralistabout PA but a pluralist about set theory and one might be anon-pluralist about ZFC and a pluralist about large cardinal axiomsand statements like CH.