In mathematical proofs, which of the following are true? I : A theorem cannot be proved by example. II: Quantifiers are important. III: Never assume any hypothesis that is not explicitly stated in the theorem. A. I only B. III only C. I, II, III D. ...
If one really wants to deal with multiple values of objects simultaneously, one is encouraged to use the language of set theory and/or logical quantifiers to do so. However, the ability to allow expressions to become only partially specified is undeniably convenient, and also rather intuitive....
But there is an alternate approach to analysis, namely nonstandard analysis, which rearranges the foundations so that many of quantifiers (particularly the existential ones) are concealed from view (usually via the device of ultrafilters). This makes the subject of analysis considerably more “algeb...
There’s a lot more to say about this subject, such as the role of intuitionistic vs. classical logic, and the relationship between quantifiers and dependent types …however, I still don’t understand these deeper aspects of the subject very well, so I won’t say any more in this post....
So, Mean, Median, Mode - what exactly are they and how are they different from one another? We hear about them a lot, and three of them feel somehow related. However, each has a distinct role in helping us make sense of data sets and numbers more easily. While we won’t go into ...
[…] elimination of quantifiers in algebraically closed fields, the existence of which follows from Hilbert’s nullstellensatz) that a set is definable if and only if it is the union of a finite number of disjoint […] Reply 22 February, 2014 at 11:07 am Jack For , one can explicitly...
But there is an alternate approach to analysis, namely nonstandard analysis, which rearranges the foundations so that many of quantifiers (particularly the existential ones) are concealed from view (usually via the device of ultrafilters). This makes the subject of analysis considerably more “algeb...
Remark 1 Another type of correspondence between hard analysis and soft analysis, which is “syntactical” rather than “semantical” in nature, arises by taking the proofs of a soft analysis result, and translating such a qualitative proof somehow (e.g. by carefully manipulating quantifiers) into...
which tends to interfere with the smooth application of algebraic laws (which are optimized for the universal quantifier rather than the existential quantifier). But there is an alternate approach to analysis, namely nonstandard analysis, which rearranges the foundations so that many of quantifiers ...