Hi, Tommy 1729!

Dear Tommy1729,

Hi again!

Concerning the supposed inconsistency of zeration, due to the non-validity of the distributive property at the ZER-PLUS levels, because it is definable via higher rank hyperoperations, I should like to start by clarifying that there seems to be a peculiar double misunderstanding in your above-mentioned example.

In fact, I suppose by mis-attention, you have mentioned a correct relationship, together with a wrong one, such as:

a * (b + c) = a*b + a*c correct!

where the stronger operation ‘distributes’ over the weaker, as well as a wrong ‘equality’ such as:

a [0] (b + c) = a [0] b + a [0]c, impossible, (first misunderstanding) because [0] is weaker than [+].

Let us see how this works in the framework of the ‘Field’ of ‘Real Numbers’ and, therefore, at the 0 - 3 ranks of the HOH (Hyper-Operations Hierarchical family). Actually, let us start at the 1 / 2 ranks (the PLUS-TIME level), where we have:

a * (b + c) = a*b + a*c, correct, because TIMES (stronger) distributes over PLUS, but:

a + b*c >< (a + b) * (a +c) the equality would be wrong, because PLUS is weaker.

[It would also do it, but in the idempotent Boolean Algebra, not in the ‘Real Field’].

Considering, now, the 0 / 1 ranks (the ZER-PLUS level), we have (sorry for the brackets), with [0]=o:

a + (b o c) = (a + b) o (a + c) correct, PLUS ‘dominates’ ZER, but:

a o (b + c) >< (a o b) + (a o c) the equality would be wrong, because ZER is ‘dominated’.

Example - let’s put: a = 2, b = 4, c = 7, we get:

2 + (4 o 7) = 2 + 8 = 10; and: (2 + 4) o (2 + 7) = 6 o 9 = 10, etc., but:

2 o (4 + 7) = 2 o 11 = 12 and: (2 o 4) + (2 o 7) = 5 + 8 = 13, !!!!!!!!

It is also useful to notice that, at the 2 / 3 ranks (the TIMES-POW level), we have:

a ^ (b * c) >< (a ^ b) * (b ^ c) and:

a * (b ^ c) >< (a * b) ^ (a * c) both wrong equalities, because the ‘level’ is not distributive.

[In this case, for levels including ranks s > 2, the distributivity does not hold, even if the operations are defined, or described, by lower rank hyperoperations]. Second misunderstanding.

Having said so, we may see that the PLUS-ZER distributivity holds, even if this would not be absolutely necessary to consider zeration as consistent, because (anyway) it doesn’hold at higher ranks.

However, there is a problem about considering -oo (minus infinite) as the ‘unit element’ of zeration (like it also happens in the ‘tropical’ Max-Plus algebra), just because -oo is NaN (Not a Number) and it does not belong to the set of the ‘Reals’. We neeed to define the actual oo, different from the oo of the classical ‘Limit Theory’ of Mathematical Analysis.

By the way, nobody ever said that there is zeration 'inside' Max-Plus algebra, but it is easy to see that the 'maximation' Max-Plus operation is a fundamental component of the RR-zeration definition.

Thank you for your interest and your kind attention.

Best regards.

(10/01/2014, 08:40 AM)tommy1729 Wrote: But with zeration that is A PROBLEM.

zeration tries to define itself in terms of HIGHER operators.

There are NO lower operators defined than zeration so its seems necc.

Any attempt at trying to define zeration with lower hyperoperations , requires those lowers to be defined as well by even lower hyperoperations ?

That would PROBABLY give an infinite descent problem.

However properties like distributive require use of a lower hyperoperator : a * (b+c) = a*b + a*c.

UNLESS , and now we are getting at the heart of the post ,

a $0$ (b + c) = a $0$ b + a $0$ c.

However after many attempts that also seems to lead to paradoxes.

tommy1729

Dear Tommy1729,

Hi again!

Concerning the supposed inconsistency of zeration, due to the non-validity of the distributive property at the ZER-PLUS levels, because it is definable via higher rank hyperoperations, I should like to start by clarifying that there seems to be a peculiar double misunderstanding in your above-mentioned example.

In fact, I suppose by mis-attention, you have mentioned a correct relationship, together with a wrong one, such as:

a * (b + c) = a*b + a*c correct!

where the stronger operation ‘distributes’ over the weaker, as well as a wrong ‘equality’ such as:

a [0] (b + c) = a [0] b + a [0]c, impossible, (first misunderstanding) because [0] is weaker than [+].

Let us see how this works in the framework of the ‘Field’ of ‘Real Numbers’ and, therefore, at the 0 - 3 ranks of the HOH (Hyper-Operations Hierarchical family). Actually, let us start at the 1 / 2 ranks (the PLUS-TIME level), where we have:

a * (b + c) = a*b + a*c, correct, because TIMES (stronger) distributes over PLUS, but:

a + b*c >< (a + b) * (a +c) the equality would be wrong, because PLUS is weaker.

[It would also do it, but in the idempotent Boolean Algebra, not in the ‘Real Field’].

Considering, now, the 0 / 1 ranks (the ZER-PLUS level), we have (sorry for the brackets), with [0]=o:

a + (b o c) = (a + b) o (a + c) correct, PLUS ‘dominates’ ZER, but:

a o (b + c) >< (a o b) + (a o c) the equality would be wrong, because ZER is ‘dominated’.

Example - let’s put: a = 2, b = 4, c = 7, we get:

2 + (4 o 7) = 2 + 8 = 10; and: (2 + 4) o (2 + 7) = 6 o 9 = 10, etc., but:

2 o (4 + 7) = 2 o 11 = 12 and: (2 o 4) + (2 o 7) = 5 + 8 = 13, !!!!!!!!

It is also useful to notice that, at the 2 / 3 ranks (the TIMES-POW level), we have:

a ^ (b * c) >< (a ^ b) * (b ^ c) and:

a * (b ^ c) >< (a * b) ^ (a * c) both wrong equalities, because the ‘level’ is not distributive.

[In this case, for levels including ranks s > 2, the distributivity does not hold, even if the operations are defined, or described, by lower rank hyperoperations]. Second misunderstanding.

Having said so, we may see that the PLUS-ZER distributivity holds, even if this would not be absolutely necessary to consider zeration as consistent, because (anyway) it doesn’hold at higher ranks.

However, there is a problem about considering -oo (minus infinite) as the ‘unit element’ of zeration (like it also happens in the ‘tropical’ Max-Plus algebra), just because -oo is NaN (Not a Number) and it does not belong to the set of the ‘Reals’. We neeed to define the actual oo, different from the oo of the classical ‘Limit Theory’ of Mathematical Analysis.

By the way, nobody ever said that there is zeration 'inside' Max-Plus algebra, but it is easy to see that the 'maximation' Max-Plus operation is a fundamental component of the RR-zeration definition.

Thank you for your interest and your kind attention.

Best regards.