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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

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
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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