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0^0

askamathematician.com

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Re: 0^0

#121

Post by chacham15 »

rickhanlonii wrote: post #86

It's very important to note here that 0^0=1 is a shorthand and not a truth . Mathematicians are absolutely not stating that they have proven, or that it is true, that 0^0=1. It is a definition, not a claim of equality. They're not saying "0^0 is 1" in the sense that they say "1+1 is 2" or "0.999... is 1". They're saying "we define 0^0 to be 1". The difference is more than just pedantry, it strikes at the core of why…

> If you do not accept it as true (an explicitly accept it as false), then you can prove all of Hyperbolic Geometry

What is the equivalent in this analogy if you do not accept that 0^0=1 (i.e. accept that 0^0=0)?

Re: 0^0

#122

Post by chunky1994 »

forrestthewoods wrote: post #117
[deleted] wrote: post #74

Earlier quoted context omitted.

[deleted]

I disagree. There are math concepts that I've found impossible to pick up from wikipedia but can easily be learned in 5 minutes by having a conversation with someone who already understands them. Honestly this article in question is a perfect example. I'd wager than almost everyone who reads Hacker News can read the blog post and understand every single step from start to finish. The same can not be said for the Wiki…

For questions about mathematical correctness I highly prefer wikipedia. To me it's far more specific and clear, while the original article makes rather short work of proofs.

When it comes down to questions that involve first principles, I would much rather have rigor over a less time consuming but potentially wrong understanding of the material, and wikipedia does quite well in providing that rigor.

Re: 0^0

#123

Post by Totient »

"...they boil down to that choice being more useful than the alternative choices, leading to simpler theorems, or feeling more “natural” to mathematicians."

Along these lines, my preferred definition of the natural numbers" is "the set of all the positive integers and, when convenient, 0".

Re: 0^0

#124

Post by Bahamut »

mathgrad wrote: post #114
rickhanlonii wrote: post #86

It's very important to note here that 0^0=1 is a shorthand and not a truth . Mathematicians are absolutely not stating that they have proven, or that it is true, that 0^0=1. It is a definition, not a claim of equality. They're not saying "0^0 is 1" in the sense that they say "1+1 is 2" or "0.999... is 1". They're saying "we define 0^0 to be 1". The difference is more than just pedantry, it strikes at the core of why…

This is not true. A very natural way to define a^b is as the number of functions for a set with "b" elements to a set with "a" elements. In this case there is exactly one function from the empty set to the empty set, and we have proven 0^0 = 1. This is no different than defining addition and then proving 1+1 = 2.

There is a bit of conflation going on here - the notation a^b being discussed is exponentiation of real numbers (or complex if you'd like to extend it that far). Of course, that notation is what inspired a^b being used to denote the number of maps from a set with cardinality b to a set with cardinality a (and the notation A^B to denote the set of all maps from B to A), but set theory was developed centuries after the likes of calculus and other important developments that touch on the reasoning used in the article linked in the OP.

Re: 0^0

#125

Post by j2kun »

joliv wrote: post #14
gweinberg wrote: post #10

Missing Q and A: But if mathematicians insist it is 1, why do high school teachers act like they know more than the mathematicians do? A: They don't. The statement that mathematicians uniformly say it is 1 is simply false. My high school teacher had a PhD in math, I think it's fair to say she was a mathematician. And yes, she said it was undefined.

I think it might be better to say it's indeterminant.

You may be confusing terms from calculus, where the label "indeterminate form" is given to 0^0 when encountered in a limit.

Re: 0^0

#126

Post by j2kun »

gweinberg wrote: post #10

Missing Q and A: But if mathematicians insist it is 1, why do high school teachers act like they know more than the mathematicians do? A: They don't. The statement that mathematicians uniformly say it is 1 is simply false. My high school teacher had a PhD in math, I think it's fair to say she was a mathematician. And yes, she said it was undefined.

Often teachers (esp high school teachers) don't have the time to give the same kind of in depth analysis of the tradeoffs.

Re: 0^0

#127

Post by chacham15 »

mathgrad wrote: post #114
rickhanlonii wrote: post #86

It's very important to note here that 0^0=1 is a shorthand and not a truth . Mathematicians are absolutely not stating that they have proven, or that it is true, that 0^0=1. It is a definition, not a claim of equality. They're not saying "0^0 is 1" in the sense that they say "1+1 is 2" or "0.999... is 1". They're saying "we define 0^0 to be 1". The difference is more than just pedantry, it strikes at the core of why…

This is not true. A very natural way to define a^b is as the number of functions for a set with "b" elements to a set with "a" elements. In this case there is exactly one function from the empty set to the empty set, and we have proven 0^0 = 1. This is no different than defining addition and then proving 1+1 = 2.

Actually this gave me the idea: a^b is the number of mappings from set a to set b. Now, if set a is empty and set b is too, it is natural to believe that there are no mappings because there is nothing to map from or to.

Re: 0^0

#128

Post by mathgrad »

j2kun wrote: post #119
mathgrad wrote: post #114

Earlier quoted context omitted.

This is not true. A very natural way to define a^b is as the number of functions for a set with "b" elements to a set with "a" elements. In this case there is exactly one function from the empty set to the empty set, and we have proven 0^0 = 1. This is no different than defining addition and then proving 1+1 = 2.

You're just pushing the arbitrariness of defining things one step further to the definition of a number. It makes it no less arbitrary that you've defined it and force us to accept the definition to get to your conclusion.

There is a very natural definition of numbers as sets. We define 0 to be the empty set and we define the successor function by S(x) = {x} union x. Then the natural numbers are the smallest set containing 0 and closed under the successor operation. This is the standard way to define the natural numbers within ZFC set theory.

This is admittedly very formal and not how the lay person thinks of natural numbers. However, it gets to the point of that although we may lie and say 0^0 = 1 is just a convention it is in fact a theorem within the system we work.

Re: 0^0

#129

Post by mathgrad »

chacham15 wrote: post #127
mathgrad wrote: post #114

Earlier quoted context omitted.

This is not true. A very natural way to define a^b is as the number of functions for a set with "b" elements to a set with "a" elements. In this case there is exactly one function from the empty set to the empty set, and we have proven 0^0 = 1. This is no different than defining addition and then proving 1+1 = 2.

Actually this gave me the idea: a^b is the number of mappings from set a to set b. Now, if set a is empty and set b is too, it is natural to believe that there are no mappings because there is nothing to map from or to.

A function is a relation for elements in a to elements in b such that for every element in a there is a unique element in b. This is vacuously true of the empty relation when a is empty.

Re: 0^0

#130

Post by impendia »

mathgrad wrote: post #114
rickhanlonii wrote: post #86

It's very important to note here that 0^0=1 is a shorthand and not a truth . Mathematicians are absolutely not stating that they have proven, or that it is true, that 0^0=1. It is a definition, not a claim of equality. They're not saying "0^0 is 1" in the sense that they say "1+1 is 2" or "0.999... is 1". They're saying "we define 0^0 to be 1". The difference is more than just pedantry, it strikes at the core of why…

This is not true. A very natural way to define a^b is as the number of functions for a set with "b" elements to a set with "a" elements. In this case there is exactly one function from the empty set to the empty set, and we have proven 0^0 = 1. This is no different than defining addition and then proving 1+1 = 2.

I think it's a little bit different. There are many natural ways of defining the expression a^b, which coincide when a and b are (say) positive integers, but there is basically only one way of defining 2 (namely, 1 + 1).
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