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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

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.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket 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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