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[Richard W]

You couldn't say that 1/3 = 0.3̅, or that 0.9̅ = 1.

Recurring decimals are simply a notation for the addition of a multiple of the reciprocal of 10^m × 10ⁿ⁻¹, so
0.3̅ is 3∕9, 0.9̅ is 9∕9 and 0.16̄ is 0.1 + 6∕90.

You're not getting Raphael's idea, which is to posit a highest number, set at one googleplex. If there is no infinity, there is no infinite recursion.

1/3 in the current system:

/-------- a googolplex of 3's -------\
.333333333333...33333333333333333333333333....

In the no-infinities system,1/3 is

/-------- a googolplex of 3's -------\
.333333333333...333333333333

These are not the same number.

Indeed, they're not. The process of basically continuing indefinitely is not a valid algorithm in the limited system. What were you going to get for 2∕3? Actually, continuing indefinitely does not work in any practical system - you never complete the calculation! (This is one reason for working with representations as (numerator, denominator) for exact calculations.) To use decimals, one has to modify normal long division to recognise when the digit sequence will repeat. Of course, sometimes the sequence will be too long - the division is then not valid in the limited system.

For addition and subtraction of recurring decimals, one would face insuperable problems when lowest common multiples don't exist.

It's true that Raphael's system would allow a finite number of rational numbers— you can still write and calculate with1/3 etc.

This still leads to problems. The smallest fraction is 1/googolplex of the [0,1] interval. But consider the number n which is 1 less than googolplex. What is 1/n? It's a division of the [0,1] interval but not the same division, and it's impossible to measure 1/n in terms of 1/googolplexes.

We could definite both in terms of 1 / (googolplex * n). But this is heresy in the no-infinity system, since you cannot have numbers above googolplex, and there is no fraction smaller than 1/googolplex.

Basically every mathematical operation fails at some point and has to be limited.

Yes, they can just fail. There is no requirement to be able to add any two numbers. For the obvious analogy, consider ordinary 32-bit addition without overflow. For many pairs (about 50% of positive integers), addition just fails.

Rather than multiplying two numbers, you can only multiply if the result is non-heretical. (This is rather like simple computer representations, in fact: you get overflow problems.)

Exactly - not every pair of numbers has a product in the limited system. The elementary binary operations are all partial operations, just as division is with unlimited integers. Subtraction also has failures if working only with natural numbers.

One more fun deduction: I said you could only have a finite number of rationals. What is that number? Both numerator and denominator can vary from 1 to googolplex, so the number is googolplex * googolplex which is forbidden.

An existing analogy is the set of ordinal numbers. Under the usual (ZF) set axioms, there's no such thing. At least, it's not a set. Possibly limiting integers should result in sets without cardinals. You missed a related but more basic issue - how many signed integers are there? (Conventionally, signed integers are the equivalence classes of pairs of natural numbers.) Even more basic - how many unsigned numbers are there? googolplex + 1 is not a valid answer!

[zompist]

You missed a related but more basic issue - how many signed integers are there? (Conventionally, signed integers are the equivalence classes of pairs of natural numbers.) Even more basic - how many unsigned numbers are there? googolplex + 1 is not a valid answer!

Good point! If someone wants to use this for a conworld, I'd suggest that there are two approaches.

1. Declare that the negative sign just defines an entire new class of numbers, so they can go from 1 to googolplex in both directions. This has the disturbing corollary that the total number of integers is heretical. But as we've seen, heresy lurks everywhere in googolplexian arithmetic.
2. Allow the number line to only go out to googolplex/2 in both directions. Not much practical difference, but how would you even state the rules, since this makes googolplex itself a forbidden number?

[Travis B.]

Even the ultrafinitists equivocate about whether there is specifically a largest number, because so much just does not work if they actually hold firm on that the number line ends somewhere.

[Richard W]

2. Allow the number line to only go out to googolplex/2 in both directions. Not much practical difference, but how would you even state the rules, since this makes googolplex itself a forbidden number?

Go back a step. Don't use the concept googolplex, but write instead 5^googol × 2^{(googol − 1)}. (I decline to expand 'googol' because I don't know the markup needed.)

[zompist]

2. Allow the number line to only go out to googolplex/2 in both directions. Not much practical difference, but how would you even state the rules, since this makes googolplex itself a forbidden number?

Go back a step. Don't use the concept googolplex, but write instead 5^googol × 2^{(googol − 1)}. (I decline to expand 'googol' because I don't know the markup needed.)

That works of course.

It makes me wonder, though, if the googolplexers could temporarily use numbers higher than googolplex (or, here, googolplex/2), much as in using Feynman diagrams you can and must use virtual particles, but they're not supposed to "really exist".

[bradrn]

It makes me wonder, though, if the googolplexers could temporarily use numbers higher than googolplex (or, here, googolplex/2), much as in using Feynman diagrams you can and must use virtual particles, but they're not supposed to "really exist".

That’s basically how people approached imaginary numbers before they were fully understood.

[Raphael]

Once again, thank you all for that very interesting discussion.

[Raphael]

Does anyone here know anything about the how and why of that apparent cultural difference where it seems like, in some parts of Europe, walking aids for frail people usually have wheels, while in the USA, they usually haven't?

[Raphael]

So even the Good Lord Himself is unimpressed with Nicholas Cage's acting career?

https://www.bbc.com/news/articles/c790l73ryp3o

[Ares Land]

Does anyone here know anything about the how and why of that apparent cultural difference where it seems like, in some parts of Europe, walking aids for frail people usually have wheels, while in the USA, they usually haven't?

I think it's not cultural, but a matter of different uses. At my mom's nursing home, they're mostly without wheels. That's because the wheels are bad for balance, but more comfortable if you have to walk some distance (which isn't the case in nursing homes.)

[Travis B.]

One thing you'll see with old peoples' walking aids here in the US is that often people will cut up tennis balls and stick them on bottoms of the legs, to enable them to slide on the ground without wearing down the rubber feet of the walkers.

[Ares Land]

One thing you'll see with old peoples' walking aids here in the US is that often people will cut up tennis balls and stick them on bottoms of the legs, to enable them to slide on the ground without wearing down the rubber feet of the walkers.

Ah, yeah, they do that too here.

[malloc]

I was right all along to worry about automation (in spirit if not necessarily in details) and everyone else was wrong.

Yes, well done indeed. Now that your sense of self-superiority is assuaged, could you please resist the temptation to make any more such remarks, and try to confine yourself to constructive contributions to the discussion, as others have been doing?

This is not about self-superiority but merely vindication for pointing out problems years before anyone else acknowledged them. Everyone dismissed me as a paranoid fool only for my predictions to come true. Quite honestly, I feel like I'm owed some apologies for all the misplaced antagonism I faced.

[bradrn]

I was right all along to worry about automation (in spirit if not necessarily in details) and everyone else was wrong.

Yes, well done indeed. Now that your sense of self-superiority is assuaged, could you please resist the temptation to make any more such remarks, and try to confine yourself to constructive contributions to the discussion, as others have been doing?

This is not about self-superiority but merely vindication for pointing out problems years before anyone else acknowledged them. Everyone dismissed me as a paranoid fool only for my predictions to come true. Quite honestly, I feel like I'm owed some apologies for all the misplaced antagonism I faced.

I mean, you did literally say ‘I was right and everyone else was wrong’. Such a statement is totally unhelpful for all purposes except making yourself feel good.

Also, I mean… not all your predictions have come true. Actually, most of your really dire ones haven’t. Even in that post, where you say ‘jobs are evaporating like steam’ — certainly it‘s a huge risk, but it hasn’t quite happened yet, I’d say. So do try to have a bit more humility.

[malloc]

Also, I mean… not all your predictions have come true. Actually, most of your really dire ones haven’t. Even in that post, where you say ‘jobs are evaporating like steam’ — certainly it‘s a huge risk, but it hasn’t quite happened yet, I’d say. So do try to have a bit more humility.

Perhaps the tech industry should also have some humility instead of imposing radical technologies designed to kick the human race out of intellectual work. If anyone merits the charge of arrogance, it's people like Sam Altman trying to create a literal Deus ex Machina and build a business around it.

[bradrn]

Perhaps the tech industry should also have some humility instead of imposing radical technologies designed to kick the human race out of intellectual work. If anyone merits the charge of arrogance, it's people like Sam Altman trying to create a literal Deus ex Machina and build a business around it.

I suspect that everyone here would heartily agree with that!

[malloc]

Having said all that, I understand that my polemics against AI have flooded the forum to a remarkable extent, with no less than three threads currently focused on them in some way. I will certainly try to find something else to talk about at least. Nonetheless, it's hard to overstate how bleak the current situation looks from my perspective. I have serious doubts about whether I even want to live in the future that is rapidly approaching. That might sound melodramatic, but I cannot stress enough how much my quality of life has deteriorated over the past few years and indeed how much blame for that falls on the tech industry in one way or another.

[alice]

@malloc: here I have to agree with you in at least one instance. When I lost my job last year and began looking for another one, so much of what was available for me was low-paid AI-model training jobs that I was glad when I qualified for benefits that I didn't have to trudge through all the slop any more.

[malloc]

Your analysis is not wrong but your conclusions are. Economic and political systems can be changed; capitalism is not a natural law, nor is an economy based on fossil fuels or nuclear power. And there are movements which try to change the world to the better; they grow stronger year after year, even in the United States, even if it currently seems as if the reactionaries were stronger. But that's politics; this thread is for discussing AI technology - and understanding the technology is fundamental to assessing the risks.

Well yes, but nothing in my post characterized capitalism or fossil fuels as unavoidable natural laws. My post clearly acknowledges the possibility of alternative economic systems. Nonetheless many aspects of technology are inherent to its mechanics and functionality and simply changing the economy will not necessarily change that. Fossil fuels would still cause global warming even under socialism and the only way to avoid their negative effects is leaving them in the ground.

For that matter, the line between politics and technology is not so clear cut. Technology impacts politics by upending the economy, creating new methods of social control, and so forth, and its introduction reflects political interests. If the tech industry succeeds in driving humans out of arts and academia (both famously left-leaning sectors), that will definitely change the political landscape.

[WeepingElf]

Your analysis is not wrong but your conclusions are. Economic and political systems can be changed; capitalism is not a natural law, nor is an economy based on fossil fuels or nuclear power. And there are movements which try to change the world to the better; they grow stronger year after year, even in the United States, even if it currently seems as if the reactionaries were stronger. But that's politics; this thread is for discussing AI technology - and understanding the technology is fundamental to assessing the risks.

Well yes, but nothing in my post characterized capitalism or fossil fuels as unavoidable natural laws. My post clearly acknowledges the possibility of alternative economic systems. Nonetheless many aspects of technology are inherent to its mechanics and functionality and simply changing the economy will not necessarily change that. Fossil fuels would still cause global warming even under socialism and the only way to avoid their negative effects is leaving them in the ground.

Indeed.

For that matter, the line between politics and technology is not so clear cut. Technology impacts politics by upending the economy, creating new methods of social control, and so forth, and its introduction reflects political interests. If the tech industry succeeds in driving humans out of arts and academia (both famously left-leaning sectors), that will definitely change the political landscape.

It would - if the people let the techbros drive them out.