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Different infinities problemby sludger13
Tags: infinities 
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#19
Jun1014, 04:21 PM

P: 75

[itex]\frac{\varepsilon^{2}}{\varepsilon}[/itex] = infinitesimal, because = [itex]\varepsilon[/itex] are only predicted from the behavior of real values (basically we don't know whether this relation works with infinitesimal values (and it's irrelevant)  those relations are postulated from [itex]\mathbb{R}[/itex]). No problem with that [itex]\mathbb{R}^{*}[/itex] definition, it is probably the easiest way. I can just define infinitesimals another way. I can probably every value from different infinity ([itex]f(x)[/itex],[itex]\frac{\mathrm{d} f(x)}{\mathrm{d} x}[/itex],[itex]\frac{\mathrm{d^{2}} f(x)}{\mathrm{d} x^{2}}[/itex]...) consider on different [itex]\mathbb{R}[/itex] number axis. I just cannot make any comparisons of those axes (I cannot compare them as two real, equal axes). The assumption is that all values from one axis are just [itex]0[/itex] in higher axis. As for [itex]\frac{\varepsilon^{2}}{\varepsilon}=\varepsilon[/itex], I'm doing exactly the same. I just postulate "miraculous transition" of [itex]\varepsilon[/itex] value to another number axis (let's say it's decreasing from higher axis). Then I divide real numbers. Also this theory is quite similar to [itex]\mathbb{R}^{*}[/itex] (if it is not the same). The entire graph (like in my first post) corresponds to one real axis. All single values from other lower axes are then [itex]0[/itex] in that graph. The advantage of this theory for me is, that I can nicely imagine the impossibility of transition from one number axis (from one infinity) to another number axis (to another infinity). As for the slope of tangent in the graph  the problem would be solved if I defined the slope in the graph as [itex]\int (\frac{\mathrm{d} f(x_{1})}{\mathrm{d} x_{1}}=0)dx[/itex]. This integral expresses every tangent slope ([itex]C(int)\in \mathbb{R}[/itex]  some solution of integral expresses every tangent). I haven't redefined much, much other stuff so far. I shall see later whether this theory would be sufficiently consistent and transparent for me. 


#20
Jun1014, 04:47 PM

P: 597




#21
Jun1014, 06:51 PM

Sci Advisor
P: 834

There are two ways to obtain infinitesimals
Serious question, when your teacher showed you the definition, didn't you do any exercises where you calculate the derivative using first principles? More to the point, you have severe misunderstandings about what is going on. If you want to know why we use the reals to do calculus with, it is because the reals are Dedekindcomplete (that means: any subset of the reals which is bounded above has a least upper bound). 


#22
Jun1114, 06:37 AM

P: 75

##\varepsilon =infinitesimal,\frac{\varepsilon^{2} }{\varepsilon}=\varepsilon## 


#23
Jun1114, 07:10 AM

P: 957

The real numbers are convenient to use in our models because they have enough range and precision to match any conceivable measurement and because as a number system they are closed under the kinds of calculations that we want to perform with them. [In practice, we may actually use IEEE floating point because we can then fit descriptions of the model numbers into a computer and because we have an entire discipline within mathematics that can deal with the quantization errors that result from this departure from the ideal models] 


#24
Jun1114, 07:37 AM

P: 75

There is an abstract meaning (either created in my brain or even "deeper" existing in physical reality) that those apples are  what we call  two. I consider that meaning as some feeling, or intuition in my brain. 


#25
Jun1114, 07:42 AM

Mentor
P: 18,299

Second, one could even argue that the natural numbers aren't so special. There are some "primitive" tribes which only recognize the numbers 1,2,3, many. Something like 4 and 6 are not distinguished. This is very much like animals see numbers. Also, this is getting dangerously in the realm of philosophy, something that this not permitted in the math sections. But I will allow the discussion for now. 


#26
Jun1114, 06:50 PM

Sci Advisor
P: 834

Take "honor". There is no arrangement of matter that lets you define precisely what honor is, and yet, (people claim) it is real. The key is the ability to communicate what honor is, i.e. you can define it with words, signs, logic, whatever. This concept will come back below. As for the "rearrange matter" definition, that's horrible when dealing with math. How do you represent the infinite dimensional vector spaces? You only have 3 dimensions to work with. What about the BanachTaski theorem? Question: Find the derivative at x=1 of the function ##f(x) = x^2## using first principles. Answer: Let ##h## be a real number satisfying ##h\neq0##. Then [tex]f(1+h)f(1) = (1+h)^21^2 = 1 + 2h + h^2  1 = 2h+h^2[/tex] [tex]\frac{f(1+h)f(1)}{h} = \frac{2h+h^2}{h} = 2+h[/tex] Then taking the limit [tex]\lim_{h\to0} \frac{f(1+h)f(1)}{h} = \lim_{h\to0}2+h = 2.[/tex] QED Notice there are no infinitesimals anywhere. π is an irrational number. This means that it's decimal (or binary or any radix you want) representation cannot be written using finite symbols. But that's not a problem, because we can calculate the decimal representation to any degree of precision we want. In fact with hexadecimal representation the BBP formula allows us to calculate the n^{th} digit without calculating the first n1 digits. Real numbers that have an algorithm to calculate the their representation are called computable real numbers. And now we have Chaitin's constant. It's real but doesn't have such an algorithm. No problem we can still define this constant using a finite string of logical symbols. This set of real numbers are the definable real numbers. Here's the kicker: the definable reals are countable. I'm assuming you know that reals are uncountable, hence the majority of real number cannot be defined. We know they exist, somehow, but no formula, equation, sentence, paragraph, arrangement of matter, or whatever will allow us to say "this is what this number is". And that's why physical reality is bad justification for the real numbers. 


#27
Jul1814, 11:31 AM

P: 75

Also an honor, like everything else, can be precisely defined by some brain structure. However I do not claim people can make that definition at this time. As for infinitedimensional vectors, man can not obviously do that. In my opinion, the point is whether one feels that IDVS is sufficiently defined. Then it's clear what am I thinking of, although it cannot be directly imagined (unlike finitedimensional vectors). Man can define IDVS using his (intuition, logical thinking...). By that definition, the concept of IDVS is uniquely specified. Also that definition represents the abstract meaning of IDVS. When I recall that definition = another brain structure... So far I couldn't think of better answer. with differentiable function, ##0## has always meaning of (what I call) lower infinity (it represents lower inf. in higher inf.). It never has the meaning of nothing. Then this is for me the proof, that I can express the derivatives in higher infinities. That corresponds to the real slope of tangent in graph. Also just one real number axis describes the graph. In differential equations, I consider all derivatives in their highest infinity, that infinity I consider as real. 


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