It does exist. The other poster just clearly showed that it exists by referring to it.
The problem is that if we include such a number in our formal system of math, we quickly find contradictions and the whole system falls apart. So such a number is incompatible with any formal system of math (though I guess you could start building one which does include such a number and see what properties it has).
Herein lies the problem, the people you are talking with do not use a form system. There system of math has something similar to the same flaw of their system of grouping of things, which would include the whole grouping that contains every grouping that doesn't contain itself. People rarely deal in formal systems and thus they can handle completely illogical statements fine as long they are protected from seeing the consequence of it.
You are certainly correct that people arguing the opposite side probably don't have a formal system in mind, but I think the intuition that an open interval in the Reals doesn't have a smallest number is easy to grasp even without any formal training. So you can force them to see the consequences of it through fairly straightforward logical contradictions.
Assume x is the smallest real number greater than 0. Then x/2 is also a real number and is greater than 0 but less than x. Therefore, x can't be the smallest real number greater than 0.
In math, when assuming the existence of something proved a contradiction, we conclude that the thing does not exist. The description may exist "integer between 3 and 4", but there is not described object. A description names a set or a class, and that class can have 0,1, or more numbers.
>In math, when assuming the existence of something proved a contradiction, we conclude that the thing does not exist.
Well only to the extent that you don't want to throw away any of the other axioms. Sometimes you do and there are some fun systems of math, but few have any practicality and those that do are often so advanced that even someone with an undergraduate focus in math can't appreciate those systems.
It is much the same with computer science. I personally enjoyed playing around with formal concepts of computation and adding some extras to see what happens. For example, what happens to a Turing machine if part of the machine can time travel or has access to an oracle. Does this make concepts like time travel inherently contradictory to our notion of computation?
But the practicality of these exercises does not exceed their entertainment value.
If you've taken Calculus, you've already worked with math that requires the infinitesimal to exist.
It's not a value you can meaningfully write out, but you can't write out pi, e, phi, root 2, 1 / 3 in base 10, root -1, etc. "I can't write it down" isn't a particularly unique property for numbers.
> If you've taken Calculus, you've already worked with math that requires the infinitesimal to exist.
Not at all. Standard calculus uses standard real numbers, for which there is no infinitesimal. One may well speak of infinitesimals as a mental tool when building a mental model for calculus, but those infinitesimals are not actual real numbers (or a well-defined mathematical object at all - in standard calculus).
There is no smallest positive infinitesimal either. At least in theories that manage to define those rigorously. And it’s mostly a formal trick anyway; standard epsilon-delta calculus avoids them entirely.
Had you actually meaningfully studied this subject, or did you just link to a Wikipedia article you half-heartedly skimmed one day?
That's a funny way to say, "No, I think you misunderstand. I mean to say no single infinitesimal number exists. Like infinity, the concept exists, but as a literal single number, no."
Or is it? Say I'm a layman and I decide that in the system of math as I understand it, 0.000... is larger than 0. Yes, if I was going to be completely form with my own system of math I would eventually have to face the problems this introduces and resolve it, but until then I can generally adopt a self contradictory system and continue to live my life unaffected. Much like many people live their whole lives using naive set theory for their understanding of sets.
Then in your system of math 0.999... is also less than 1.
However, basic arithmetic taught to children requires that adding trailing zeros does not change the value of a number. You'll have a hard time doing arithmetic once you change that assumption.
Wildberger is great. His lectures that he teaches at UNSW (i think) are interesting, and he usually keeps a clear dividing line between std math and his own predilections. It threads the line between being a kook and legitimate published mathematician very finely.
I actually have some sympathies with his contention that real numbers (limit points of infinite series) are somehow a different animal than rational numbers. But it might be easier for me to go there because practically all numbers on computers that we work with are rational, floating point values. On the other hand, it seems like a philosophical distinction in the end because you can fully order them both on a number line.
If I give you two representatives of real numbers, say turing machines that write out on their tape the binary digits of those real numbers, in general you will not be able to order them.
Certainly not non-computable ones, but presumably they lie somewhere regardless of my inability to do it on a TM. Which presumably gives rise to all the weirdness uncountable infinities give you.
I guess I shouldn't phrase it as "you can fully order it". :D
Zermelo's theorem at that point right?
Even computable reals do not have computable ordering.
> but presumably they lie somewhere regardless of my inability to do it on a TM.
Why?
> Zermelo's theorem at that point right?
It is declared by fiat in standard set theory that infinite sets can be well ordered. This is no real mathematical justification. The real justification is social: that it is convenient for mathematicians to not care about the ontology of these nasty infinite objects so long as results are mostly reasonable for objects that mathematicians actually care about. You don't get into too much trouble pretending the reals are nice so long as you don't look too hard.
A mathematician might say that this shows that you do not really have accurate floating point values and arithmetic in your computer, but instead something close to it.
I don't see how generators would change anything here. "with effects", "stateful", and the integration between the two are all equally important in the statement.
Well generators are a good primitive to represent "stateful" "functions" "with effects". The concept of hooks, which somehow give the control back to React to do something, map marvelously to react.
I'd be surprised if generators didn't come into React at one point or another
I dunno, that's what happens in a global shortage sometimes. Remember that the governor of Maryland isn't exactly a small-time official - Maryland's economy is about as large as Iran or Norway.
So should black people stop organizing along lines of race, because that doesn't seem like an effective way to fight racism? That just means less black representation.
Yes. We shouldn't strap on blinders and ignore race, of course. But the activists who go around encouraging people to identify as Black first and individuals second, I think are toxic and ineffective.
Plenty of people are hurt or killed through clerical errors by medical professionals, but you're not going to get anyone admitting that. Even for something simple, if you have any accusation to make in a medical context, it's probably time to evaluate your options and consider a new doctor, and only then make your accusation.
I think tech moved too quickly and in some ways, due to power being left on the table by government, tech became responsible; much like how businesses have taken up the responsibility of American healthcare due to the issue being under-addressed by government.
Now that YouTube is used by people of all ages, Google has to make social and political decisions about what kind of content is appropriate for different ages.
