By Jacques Fleuriot PhD, MEng (auth.)

Sir Isaac Newton's philosophi Naturalis Principia Mathematica'(the Principia) features a prose-style mix of geometric and restrict reasoning that has frequently been considered as logically vague.

In **A blend of Geometry Theorem Proving and Nonstandard****Analysis**, Jacques Fleuriot provides a formalization of Lemmas and Propositions from the Principia utilizing a mix of equipment from geometry and nonstandard research. The mechanization of the methods, which respects a lot of Newton's unique reasoning, is built in the theorem prover Isabelle. the appliance of this framework to the mechanization of uncomplicated actual research utilizing nonstandard recommendations is additionally discussed.

**Read or Download A Combination of Geometry Theorem Proving and Nonstandard Analysis with Application to Newton’s Principia PDF**

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X n + Y n})" hypreal_Iess_def "p < (Q:: hypreal) == 3x Y. XERep_hypreal (p) A YERep_hypreal(Q) A {n: :nat. 3. 1 Choosing a Free Ultrafilter To start the construction, a free ultrafilter UIN is chosen on the set of natural numbers IN. There exists one according to the Weak Ultrafilter Theorem: (WUF) Weak Ultrafilter Theorem. There exists a free ultrafilter on IN. As can be seen, this is a special case of the UFT corollary from the last section. In fact, we have the implications AC => UFT => WUF, which are not reversible.

This result can be proved using Zorn's Lemma. In fact, we are really interested in proving a corollary of UFT about the existence of free ultrafilters: 1 1 Some authors like Hoskins (48) and Keisler [52) state the corollary (or even one of its special cases) as the actual Ultrafilter Theorem. 44 3. Constructing the Hyperreals Corollary. On every infinite set there exists a free ultrafilter. ,finite S ==> 3u. u E FreeUltrafilters S To do so, we define in the theory Filter the set, SuperFrechet S, of all filters on S that contain the Fnkhet filter (Le.

The usual approach is to arrange them in a lattice respecting the inclusions between the sets. Let 2Z,