I think it’s important to be aware of what kind of functions were working with in googology. All functions act as a map from a domain set D to a range set R. All of the functions we use here go from the natural numbers to the natural numbers (i.e. f: N→N), so they should never have an output of infinity because it isn’t a natural number. Of course in the case of this G function, you could construct a new set, N ∪ {∞}, which contains all the naturals as well as infinity, and use that as the range. However, usually we don’t do that and instead say that the function is undefined for certain values of n. If nothing else, it’s good to remember that in googology, we work with natural numbers, not infinity!
@@carbrickscity I can't come up with any way that infinity would be useful in googology, the closest is ordinals, but defining them as infinite is only productive in simplifying definitions and proofs in set theory, it doesn't actually serve a purpose here.
Infinity helps us understand big numbers in some way. And it definitely plays a role in googology whether it is useful or not. Maybe different people has their definition of useful, but to some people there is some use. Ordinal is one, cardinals is another. And don't forget people often bring up infinity when talking about big numbers. Even though infinity is not a number, but people still often thinks of infinity when we talk about big numbers. Infinity is bigger than any finite number. Then there are cardinals that are related to infinities.
@@carbrickscity what's funny is there isn't just one infinity and and infinity seem to work as a hierarchy (eg) infinite amount of decimal between natural numbers then an infinite amount of number both negative and positive then a ln infinite amount of function to either increase and/or decrease said values and so on and so forth.
I recently saw a question that went something like this: ignoring physics and even the time left in the universe, you have a decision to make, you can be put in a room in which you suffer immensely for a period of time. And after that period of time is over, you are transferred to a so-called heaven where u will be eternally happy and in peace, or you can be put into the so called heaven first, for a period of time and then be transferred into the room of suffering for the rest of eternity. Now, the time you spend in the first room, whether it be the suffer room or the heaven room will be G64 amount of years, or even TREE(3) amount of years. So even if you chose the heaven room first, you will still only have a finite amount of time, even if that time is TREE(3) amount of years until you will be sent to the suffer room for the rest of eternity. The question is, which room would u go in first? If it seems obvious to pick the heaven room because of how big TREE(3) is, what is the maximum amount of years you would spend on the suffer room first before being sent to the heaven room?
My function: FINAL(n) FINAL(0)=0 FINAL(1)=1 FINAL(2)=10 FINAL(3)=10^100 FINAL(4)=10^10^100 FINAL(5)=G64 FINAL(6)=TREE(3) FINAL(7)=SSCG(3) FINAL(8)=SCG(13) FINAL(9)=Tar(3) FINAL(10)=D^5(99) FINAL(11)=Rayo(10^100) FINAL(12)=Fish number 7 FINAL(13)=Large number garden number FINAL(14)=The smallest valid googolism that is larger than large number garden number FINAL(15)=The smallest valid googolism that is larger than FINAL(14) FINAL(16)=The smallest valid googolism that is larger than FINAL(15) ... My number: FINAL's number FINAL^FINAL(FINAL(10^10^100))(10^10^100) (FINAL(FINAL(FINAL(...(10^10^100)))...)))(FINAL(FINAL(10^10^100)) times)) This number is so large that it completely breaks math.(FINAL's number^FINAL's number=infinity)
This mathematical function seems to be undefined because the way in which each value is calculated for each entry of the function is not explained, you literally put the largest numbers in the mathematics as results of inputs in your function to show that your function is the "limit of mathematics." I don't need to say more, the joke tells itself, just seeing that you said: "FINAL Number^Final Number=Infinite" is already a source of ridicule because no finite number, making it any operation or method that focuses on the growth of a number It will never be infinite.
I think it’s important to be aware of what kind of functions were working with in googology. All functions act as a map from a domain set D to a range set R. All of the functions we use here go from the natural numbers to the natural numbers (i.e. f: N→N), so they should never have an output of infinity because it isn’t a natural number. Of course in the case of this G function, you could construct a new set, N ∪ {∞}, which contains all the naturals as well as infinity, and use that as the range. However, usually we don’t do that and instead say that the function is undefined for certain values of n.
If nothing else, it’s good to remember that in googology, we work with natural numbers, not infinity!
Yes, googology is about big numbers and infinity is not even a number. But still infinity plays a big role in googology.
Well, yes. Number to large to understand or even accurately quantify but they are natural nonetheless .
@@carbrickscity I can't come up with any way that infinity would be useful in googology, the closest is ordinals, but defining them as infinite is only productive in simplifying definitions and proofs in set theory, it doesn't actually serve a purpose here.
Infinity helps us understand big numbers in some way. And it definitely plays a role in googology whether it is useful or not. Maybe different people has their definition of useful, but to some people there is some use. Ordinal is one, cardinals is another. And don't forget people often bring up infinity when talking about big numbers. Even though infinity is not a number, but people still often thinks of infinity when we talk about big numbers. Infinity is bigger than any finite number. Then there are cardinals that are related to infinities.
@@carbrickscity what's funny is there isn't just one infinity and and infinity seem to work as a hierarchy (eg) infinite amount of decimal between natural numbers then an infinite amount of number both negative and positive then a ln infinite amount of function to either increase and/or decrease said values and so on and so forth.
I recently saw a question that went something like this: ignoring physics and even the time left in the universe, you have a decision to make, you can be put in a room in which you suffer immensely for a period of time. And after that period of time is over, you are transferred to a so-called heaven where u will be eternally happy and in peace, or you can be put into the so called heaven first, for a period of time and then be transferred into the room of suffering for the rest of eternity. Now, the time you spend in the first room, whether it be the suffer room or the heaven room will be G64 amount of years, or even TREE(3) amount of years. So even if you chose the heaven room first, you will still only have a finite amount of time, even if that time is TREE(3) amount of years until you will be sent to the suffer room for the rest of eternity. The question is, which room would u go in first? If it seems obvious to pick the heaven room because of how big TREE(3) is, what is the maximum amount of years you would spend on the suffer room first before being sent to the heaven room?
For these kind of questions I think most people would pick "suffer" first as long as the "heaven" is forever. That seems logical.
Tree 3 is insane but its still technically an infinite payoff.
YEAH@@carbrickscity
Got excited when I saw you posted a new vid!
Explain Cascading-E-notation please
I agree
Does stuff like 2^n or n! counts?
Sure.
Bet, new math video
My function: FINAL(n)
FINAL(0)=0
FINAL(1)=1
FINAL(2)=10
FINAL(3)=10^100
FINAL(4)=10^10^100
FINAL(5)=G64
FINAL(6)=TREE(3)
FINAL(7)=SSCG(3)
FINAL(8)=SCG(13)
FINAL(9)=Tar(3)
FINAL(10)=D^5(99)
FINAL(11)=Rayo(10^100)
FINAL(12)=Fish number 7
FINAL(13)=Large number garden number
FINAL(14)=The smallest valid googolism that is larger than large number garden number
FINAL(15)=The smallest valid googolism that is larger than FINAL(14)
FINAL(16)=The smallest valid googolism that is larger than FINAL(15)
...
My number: FINAL's number
FINAL^FINAL(FINAL(10^10^100))(10^10^100)
(FINAL(FINAL(FINAL(...(10^10^100)))...)))(FINAL(FINAL(10^10^100)) times))
This number is so large that it completely breaks math.(FINAL's number^FINAL's number=infinity)
Not well defined and anyone can just say your number + 1. Also no number can reach infinity by using exponents.
It's because this number is not a normal number at all.
Just saying it's a salad number nonetheless and anyone can say the same thing.
This mathematical function seems to be undefined because the way in which each value is calculated for each entry of the function is not explained, you literally put the largest numbers in the mathematics as results of inputs in your function to show that your function is the "limit of mathematics." I don't need to say more, the joke tells itself, just seeing that you said: "FINAL Number^Final Number=Infinite" is already a source of ridicule because no finite number, making it any operation or method that focuses on the growth of a number It will never be infinite.