Re: Simple enough for every reader?

On 2025-06-26 13:09:32 +0000, WM said:

On 26.06.2025 12:36, Mikko wrote:

On 2025-06-25 20:08:06 +0000, WM said:
 

On 25.06.2025 09:53, Mikko wrote:

On 2025-06-24 17:33:20 +0000, WM said:
 

On 24.06.2025 11:15, Mikko wrote:

On 2025-06-23 10:59:08 +0000, WM said:

 

The natural numbers are well-ordered, from the first to the last which is subtracted. Does this change if all can be subtracted?

 There is no last natural number. Every subset of natural numbers
has a first member (in the arithmetic order) but infinite subsets
don't have a last one.

 Then it is impossible to remove all of them in their order.

 Maybe so. There expression "to remove all of them in their order" is
an informal expression that might refer to something that can be
expressed mathematically but maybe it does not.

 You confuse mathematics with formalism. The greatest mathematicians have lived before formalism had been invented. Formalism is only a crutch for those who cannot think without crutches.

 Formalism was invented to avoid or correct the errrors the greatest
mathematicians made before formalism was invented. The invoention
of formalism was not a sudden event but a gradual process that took
thousands of years.

 Say about 100 years.

For most of the development. But it started much earlier. Formal inference
rules of inference were discussed and identified by Aristotle, who used
sigle symbols for sentence variables. Euclid had formal statements of
postulates and axioms (but not inference rules). An even earlier invention
is the use of a formal language (with a syntax different from hatural
languages) for expression of quantities.

 

If all naturals can be subtracted then this can be done in their order. For enumerating another set the order is even a precondition.

 For enumeration an order is necessary but not sufficient. But for
subtraction it is not.

 If we subtract in the order that is used for enumerating then a last one is necessary.

No, there is no last one in an infinete enumeration. The word "infinite"
originally meant "having no end". We can remove all odd numbers from the
natural numbers, leaving the even numbers, but there is no last number
removed.

For example, irrational numbers have their
arithmetic order but cannot be enumerated.

 The reason is that infinite sets cannot be enumerated. Also there completion would necessitate a last one.

Infinite enumerable sets can. Then the enumeration is an infinte sequence.

< But they can be sutracted

from the set of real numbers and the result is the set of rational
numbers.

That can be done collectively only.

Doesn't matter.
--
Mikko