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The critical point of his interpretation of a non-Euclidean geometry is this: it is embedded in Euclidean geometry! So we are able to prove (within the embedding Euclidean geometry) that the disc-postulates hold ( e.g. , we can prove the statement [*]above as a theorem about circular arcs in Euclidean geometry).Therefore, if there is any inconsistency in non-Euclidean geometry, then that could be parlayed into some inconsistency of Euclidean geometry.Thus, his interpretation gives a proof that the strange non-Euclidean geometryis as sound as our familiar Euclidean geometry.

P vs. np and oracles

A well-known problem in computer scienceP vs. NPasks whether (for a given problem) it is truly moredifficult to find a short solution (when one exists) (NP), than it is to verify a short purported solution handed to you(P). For example,Given a set of people and how strong person is, can you partition them into two tug-of-war teamswhich are exactly evenly matched?Certainly it seems easier to check that a pair of proposed rosters has equal strength(and, verify that everybody really is on one team or the other)than to have to come up with two perfectly-matched teams. But conceivably, the two tasks might be equally-difficultup to some acceptable (polynomial time) overhead. While every assumes that P is easier than NP,nobody has been able to prove it.

An interesting variant of the problem lets both the problem-solver and the purported-answer-verifier each have access toa particular oracle a program that will gives instant yes/no answers to some other problem (say,given any set of numbers, yes or no: is there an even-sized subsetwhose total is exactly the same as some odd sized subset?).

It has been shown that there is some oracle which makes theproblem-solver's job provably tougher than the proof-verifier's job, and also there is some other oracleproblem-solver's job provably no-tougher than the proof-verifier's job.

This means that any proof of P being different from NP has to be subtle enough so thatwhen P and NP are re-interpreted asP and NP with respect to a particular oracle, the proof will no longer go through.Unfortunately, this eliminates all the routine methods of proof; we know that solving this problem will take some new attack.

LWenheim-skolem and the real numbers

The Lwenheim-Skolem theorem of logic states that if a set of (countable) domain axioms has a model at all,then it has a countable model. This is a bit surprising when applied to the axioms ofarithmetic for the real numbers: even though the real numbers are uncountable,there is some countable model which meets all our (finite) axioms of the real numbers!

Object-oriented programming

Note that object-oriented programming is founded on the possibility for nonstandard interpretations:perhaps you have some code which is given a list of Object s, and you proceed to call the method toString on each of them. Certainly there is a standard interpretation for the function Object.toString , but your code is built to work even when you call this function andsome nonstandard, custom, overridden method is called instead.

It can become very difficult to reason about programs when the run-time method invoked might be different from the one being called.We're used to specifying type constratins which any interpretation must satisfy;wouldn't it be nice to specify more complicated constraints, e.g. this function returns an int which is a valid index into [some array]? And if we can describe the constraint formally (rather than in English comments, which is how most code works), then we could have the computer enforce that contract!(for every interpretation which gets executed, including non-static ones).

An obvious formal specification language is code itselfhave code which verifies pre-conditions before calling a function,and then runs code verifying the post-condition before leaving the function. Indeed,there are several such tools about ( Java , Scheme ). In the presence of inheritance, it's harder than you might initially think todo this correctly .

It is still a research goal to be able to (sometimes) optimize away such run-time verifications;this requires proving that some code is correct (at least, with respect to its post-condition).The fact that the code might call a function which will be later overridden (ournon-standard interpretations) exacerbates this difficulty.(And proving correctness in the presence of concurrency is even tougher!)

Even if not proving programs correct, being able to specify contracts in a formallanguage (code or logic) is a valuable skill.

Real-world arguments

Finally, it is worth noting that many rebuttles of real world arguments (see also some exercises ) amount to showing thatthe argument's form can't be valid since it doesn't hold under other interpretations, and thus there mustbe some unstated assumptions in the original.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Intro to logic. OpenStax CNX. Jan 29, 2008 Download for free at http://cnx.org/content/col10154/1.20
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