Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

The field of “useful reals” between rational and real numbers (2019)

chittur.dev

Post reply on HN
Print view View original

101–110 of 121 posts

Re: The field of “useful reals” between rational and real numbers (2019)

#101

Post by joppy »

lonelappde wrote: post #82

Such a weird perspective. The author thought they discovered something that true and interesting and kind of fundamental but wasn't already published, but didn't think it was worth publishing to the math community?

Thinking about maths is fun, and some people do it for leisure and write what they find in innocuous places like blogs. Usually the things you come up with are already well-known by a different name (as was the case here), so one would usually not publish something like this.

Think of it just like a random blog post on someone’s thoughts. Just because it contains maths doesn’t mean it needs to be published or not, it can be free to live its own life.

Re: The field of “useful reals” between rational and real numbers (2019)

#102

Post by voxl »

D_Alex wrote: post #84

If you look at the integers between say Graham's number ( https://en.wikipedia.org/wiki/Graham%27s_number ) and TREE3( https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem ) you can observe that practically all of these integers, while "computable", cannot be defined within the known constraints of this universe. Which raises an interesting question: In what meaningful sense do these numbers exist ? They are just o…

You're mixing up "defined" with "defined via a decimal numeral" we can define these numbers without much difficulty via finite formula that compute them. This is a completely valid definition, it is just not a decimal numeral.

An interesting idea might be "useful integers" which requires whatever definition we have to allow approximation of any finite subsequence with error converging to zero given more computational power.

Re: The field of “useful reals” between rational and real numbers (2019)

#103

Post by minikites »

circlefavshape wrote: post #95
D_Alex wrote: post #84

If you look at the integers between say Graham's number ( https://en.wikipedia.org/wiki/Graham%27s_number ) and TREE3( https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem ) you can observe that practically all of these integers, while "computable", cannot be defined within the known constraints of this universe. Which raises an interesting question: In what meaningful sense do these numbers exist ? They are just o…

In what meaningful sense do any numbers exist? This comes up with my kids sometimes ... are numbers real?

I liked this Numberphile video on that topic: https://www.youtube.com/watch?v=1EGDCh75SpQ

Re: The field of “useful reals” between rational and real numbers (2019)

#104

Post by klodolph »

Someone wrote: post #94
klodolph wrote: post #23

Earlier quoted context omitted.

Yes—and to elaborate, the reason why an algebraic field extension of ℚ cannot contain π is because: - If it is a field, it contains π, π², π³, … which are linearly independent. - By definition, an algebraic field extension is finite dimensional.

”π, π², π³, … which are linearly independent.” Do we know that? My search doesn’t get more than https://www.encyclopediaofmath.org/index.php/Lindemann_theor... , which proves it for “𝑒, 𝑒², 𝑒³, …“.

There are two ways we can go about this. We can either take a closer look at the Lindemann theorem, or we can talk about the definition of “transcendental number”. I’m not going to put a full proof here.

If you look at the Lindemann theorem, you can transform the equation so that it uses π instead of e. Multiply all of the exponents by i (which is algebraic!) and then use Euler’s identity. You end up with the same formula, but with π instead of e.

However, if we already know that π is transcendental (which is proven by the Lindemann theorem using the above technique), we can rewrite any linear combination of B = {1, π, π², π³, …} as P(π) where P is a polynomial with coefficients in ℚ. Because π is transcendental, we know that P(π)=0 only if P is the zero polynomial (that is the definition of transcendental number).

In general, one of the big tricks here is that the set of polynomials is a vector space, and the powers B = {1, x, x², x³, …} span the entire vector space.

Re: The field of “useful reals” between rational and real numbers (2019)

#105

Post by pwdisswordfish2 »

voxl wrote: post #102
D_Alex wrote: post #84

If you look at the integers between say Graham's number ( https://en.wikipedia.org/wiki/Graham%27s_number ) and TREE3( https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem ) you can observe that practically all of these integers, while "computable", cannot be defined within the known constraints of this universe. Which raises an interesting question: In what meaningful sense do these numbers exist ? They are just o…

You're mixing up "defined" with "defined via a decimal numeral" we can define these numbers without much difficulty via finite formula that compute them. This is a completely valid definition, it is just not a decimal numeral. An interesting idea might be "useful integers" which requires whatever definition we have to allow approximation of any finite subsequence with error converging to zero given more computational…

GP did not mix up anything. Some of those finite formulas also will be too long to be written within the constraints of this universe. The pigeonhole principle applies just as much to finite formulas as it does to finite strings of decimal digits.

Re: The field of “useful reals” between rational and real numbers (2019)

#106

Post by H8crilA »

steerablesafe wrote: post #96
lonelappde wrote: post #80

Earlier quoted context omitted.

Completeness in the "full" reals is a useless feature, though. All is gives you is an emotional crutch to pretend your cauchy sequences can be mapped to regular numbers. But it doesn't give you anything you didn't already have in the cauchy sequences and useful reals.

You are of course right, reals are isomorphic to equivalence classes of Cauchy sequences on Q. But once you are dealing with equivalence classes of Cauchy sequences on Q you might as well give it a name. Maybe call it R.

His point is different. You cannot (by definition) ever write a "name", a formula, a rule, a lim expression, anything really, for a real that is not in the useful reals.

Re: The field of “useful reals” between rational and real numbers (2019)

#107

Post by dwohnitmok »

dwohnitmok wrote: post #85
Sniffnoy wrote: post #35

None of this is somehow secret. The standard name for this is "definable"[0]. Although, one has to be really careful with this sort of thing; there are apparently a number of subtle logical issues[1] that come up when talking about these... (Note, by the way, that there's any number of other fields one could put inbetween; such as the field of algebraic reals, or computable reals, or the fraction field of the ring of…

Well that Math Overflow post is excellent. One of the logical issues is that there is a model of ZFC where all reals are definable/useful. I'm guessing that's not what the author of this blog post is going for... If this seems impossible given that the number of definitions is countable, note first that it is possible that a model of ZFC is itself countable (in a larger ambient model), but it cannot witness the count…

Upon re-reading my reply, it might be worthwhile to simply quote Hamkins' conclusion in full.

> And therefore neither are you able to do this in general. The claims made in both in your question and the Wikipedia page [the Wikipedia page has now since been updated] on the existence of non-definable numbers and objects, are simply unwarranted. For all you know, our set-theoretic universe is pointwise definable, and every object is uniquely specified by a property.

Re: The field of “useful reals” between rational and real numbers (2019)

#108

Post by perl4ever »

shawnz wrote: post #66
perl4ever wrote: post #63

Earlier quoted context omitted.

"It’s larger than the rational numbers in the sense that it is a strict superset" Ok, did I miss the explanation of that? Or is it something in "part 2" which I didn't see a link to?

Clearly every single rational number is "useful", plus others that are not rational

So I guess what I lack is an understanding of why that doesn't affect cardinality.

Re: The field of “useful reals” between rational and real numbers (2019)

#109

Post by doomrobo »

lonelappde wrote: post #74
jopolous wrote: post #67

Earlier quoted context omitted.

I guarantee that is not an algebraic extension. It's not even a finite extension

That's not a valid critique, as other commenters explained

Finite implies algebraic. The other comment says that the converse isn't true.

Re: The field of “useful reals” between rational and real numbers (2019)

#110

Post by PureParadigm »

dchyrdvh wrote: post #91
PureParadigm wrote: post #88

Earlier quoted context omitted.

Why is the alphabet not countable? If each time you think of a new idea and make a symbol for it, I can also assign it to an integer (because there is always a next integer like there is always a new symbol you can come up with). When you come up with a new concept, it should also be possible to write out a definition of it. If you can write down your definition (in English, math notation, etc.), then it comes from a…

We don't "come up" with ideas from other ideas using some closed form rules of logic, like in Coq or some Turing machine. Instead, we discover new ideas. There is a world of ideas and the real world. People live in both worlds. When they discover a new idea, often by accident, they label it with a symbol and use it in the real world. Other people can see the same idea and since they can't fully describe it with words…

Even if ideas come from an uncountable set (not convinced yet), there are still only countably many ideas people will ever have. Each time anyone comes up with an idea, I can assign it a new integer.
Post reply on HN