/sci/ - Science & Math

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Anonymous Thu Oct 23 15:08:29 2014 No.6830989 [Reply] [Original]

Is there any proof that infinity, infinitesimals and continuity exist? Why would continuous real numbers exist? You can't use infinity or infinitesimals to prove that they do by the way.

>>	Anonymous Thu Oct 23 15:11:05 2014 No.6830995 axioms

>>	Anonymous Thu Oct 23 15:11:23 2014 No.6830996 What the fuck are you talking about?

Anonymous Thu Oct 23 15:23:50 2014 No.6831014

Infinitesimals do not exist in the standard formulation of the reals (Dedekind cuts or Cauchy sequences of rationals). One can easily see why after considering the Axiom of Archimedes:

<span class="math">\forall x\in\mathbb{R} \exists n\in\mathbb{N} : x<\frac{1}{n} [/spoiler]

Hence, there is no smallest real.

However, there is a large field called "non-standard analysis" which does admit infinitesimals, though I'm not familiar with it.

Non standard analysis is not anymore powerful than standard analysis-- any theorem you can prove in non-standard analysis you can prove in standard analysis.

I think some prefer non-standard for philsophical reasons, but the profession generally uses standard analysis. If I'm not mistaken, one can say that the difference between them is the axiom mentioned above.

>>	Anonymous Thu Oct 23 15:26:30 2014 No.6831015 >>6831014 oops should be <span class="math">\forall x\in\mathbb{R}, \exists n\in\mathbb{N} : \frac{1}{n}< x<span class="math">[/spoiler][/spoiler]

>>	Anonymous Thu Oct 23 15:34:20 2014 No.6831026 >>6831014 Get on the nonstandard analysis hype train: https://www.youtube.com/watch?v=IS9fsr3yGLE

>>	Anonymous Thu Oct 23 15:49:40 2014 No.6831039 >>6831014 Is the axiom of Archimedes a tautology?

Anonymous Thu Oct 23 15:52:33 2014 No.6831043

>>6830989
Assume the set of naturals is finite.
this implies that there exists some natural number N such that N is greater than or equal to all natural numbers. Since there is an axiom in number theory such that for any natural number n, n+1 is also a natural number, we know that adding 1 to N creates a natural number. Since there is an axiom that states x+1 > x for all real numbers, and the naturals are a subset of the reals, we know that there is a natural number N+1 such that N+1 > N. This contradicts our assertion that N is greater than or equal to all natural numbers. Since N is arbitrarily defined, we know that this property exists for all natural numbers. Therefore, we know that there is no largest natural number. For every natural number, there is a greater natural number. Therefore the cardinality of the naturals is infinite. This proves that infinite exists as a concept describing a set of numbers

>>	Anonymous Thu Oct 23 15:54:25 2014 No.6831048 >>6831039 in standard analysis it's a theorem not an axiom. it's called an axiom for historical reasons. It follows from the definition of order on the reals

>>	Anonymous Thu Oct 23 15:59:03 2014 No.6831052 >>6831014 nigga you centuries behind infinitesimals are real

>>	Anonymous Thu Oct 23 15:59:22 2014 No.6831053 >>6831026 I can't understand what that guy says.

>>	Anonymous Thu Oct 23 16:14:29 2014 No.6831077 File: 28 KB, 150x191, math10.jpg [View same] [iqdb] [saucenao] [google] >>6831026 Why listen to this con when his whole field is debunked by this man?

>>	Anonymous Thu Oct 23 16:18:48 2014 No.6831082 >>6831077 Why listen to Wildberger when he was debunked by Jacob Barnett?

>>	Anonymous Thu Oct 23 16:36:53 2014 No.6831128 >>6831082 Why listen to Jacob Barnett when he was debunked by /sci/?

>>	Anonymous Thu Oct 23 16:37:45 2014 No.6831130 >>6831128 Because he wasn't.

>>	Anonymous Thu Oct 23 16:38:39 2014 No.6831132 >>6831128 Who listen's to /sci/ anyway?

>>	Anonymous Thu Oct 23 16:44:38 2014 No.6831150 >>6831130 Counterpoint: Yes he was.

>>	Anonymous Thu Oct 23 16:49:29 2014 No.6831170 File: 2.79 MB, 853x480, infinite decimals.webm [View same] [iqdb] [saucenao] [google]

>>	Anonymous Thu Oct 23 16:56:29 2014 No.6831188 >>6831170 that was great Wildberger 4 lyfe

>>	Anonymous Thu Oct 23 16:57:19 2014 No.6831189 >>6831170 Remember that time when people actually gave a shit about your intuitionist math? Yeah, I don't either, because everybody who was alive back then is dead now.

>>	Anonymous Thu Oct 23 17:07:42 2014 No.6831209 >>6831189 Almost anyone who writes actually-formal proofs does it some form of intuitionistic type theory. ZFC is only popular among people who don't actually use it.

>>	Anonymous Fri Oct 24 02:35:31 2014 No.6832225 >>6831014 >Infinitesimals do not exist Ok, so differentials don't exist either, and thus the derivative being the quotient of two differentials (dy/dx) must be bullshit either.

>>	Anonymous Fri Oct 24 02:38:44 2014 No.6832229 >>6831015 >for any real number x there exists a natural number n the inverse of which is smaller than x Yes, so? How does that defy the existence of infinitesimals?

Anonymous Fri Oct 24 03:08:25 2014 No.6832251

>>6831043
You don't need tu use R to prove that result, the definition of the successor over the interger is enough (Peano axioms basically). Your proof still works without asuming x+1>x for all x in R.

>>6831014
>>6831015
You Archimedes Axiom doesn't entails that there is no smallest real, you simply proved that you can find strictly positive reals as close to zero as you wish. In order to prove that there is no smallest real you must consider negative reals as well.

The difference between standard and non-standard analysis is three axioms, that are a part of non-standard theory and not a part of standard theory. The essential bonus with those axioms is that, given a set E, they allow you to deduce the existence of an object x for which P(x,A) is true for all standard A included in E as long as for all standard finite A included in E there is a standard y such that P(y,A) is true.

For instance, for all finite family of standard (traditional) integer numbers, there is an standard integer stricly greater than all members of the family. Non-standard analysis allows you to deduce from that that there is a (non-standard) integer number stricly greater than all standard integers.

You can see it as a clever trick for writing proofs faster and making problems easier.

>>	Anonymous Fri Oct 24 03:16:22 2014 No.6832258 >>6831043 This guy gets it

Anonymous Fri Oct 24 03:35:29 2014 No.6832281

>>6832251
But numbers are not actually real and math is a tool not a science. Trying to calculate the true value of an imaginary concept is just give you a proof for circle jerking.

You are all faggots, study something that exists in this fucking universe.

>>	Anonymous Fri Oct 24 03:45:09 2014 No.6832293 >>6832281 That was a terrible troll attempt. Be a little more subtle.

>>	Anonymous Fri Oct 24 05:08:26 2014 No.6832397 >>6832225 wow this guy seriously doesn't get it, Retake calc I. >>6832229 It doesn't dipshit >>6832281 >How Can Ouw NumbNumbs Gets Real Iff I's Not Reals???

>>	Anonymous Fri Oct 24 12:33:15 2014 No.6832919 >>6832251 I take it you learned nonstandard with the IST route?

>>	Anonymous Fri Oct 24 13:16:50 2014 No.6832974 >>6832281 infinity exists in the universe.

Anonymous Fri Oct 24 13:21:45 2014 No.6832983

Charles Sanders Peirce rules!
"In 1908 Peirce wrote that he found that a true continuum might have or lack such room. Jérôme Havenel (2008): "It is on May 26, 1908, that Peirce finally gave up his idea that in every continuum there is room for whatever collection of any multitude. From now on, there are different kinds of continua, which have different properties."[93]"

>>	Anonymous Fri Oct 24 13:23:55 2014 No.6832985 >>6832983 How do I Lowenheim-Skolem?

>>	Anonymous Fri Oct 24 13:29:47 2014 No.6832987 >>6832985 I find it is useful to believe in a ruler.

>>	Anonymous Fri Oct 24 14:00:16 2014 No.6833049 >>6832225 >quotient of two differentials >watch as I take a differential operator and use it as a two separate variables You're the guy in this image aren't you? >>6831236

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