Re: Readings in (some of the) foundations of mathematics --- tree of knowledge

On 6/27/2026 7:59 PM, olcott wrote:
> On 6/27/2026 6:33 PM, dbush wrote:
>> On 6/27/2026 7:27 PM, olcott wrote:
>>> On 6/27/2026 5:53 PM, dbush wrote:
>>>> On 6/27/2026 6:44 PM, olcott wrote:
>>>>> On 6/27/2026 5:33 PM, dbush wrote:
>>>>>>
>>>>>> So you're claiming the sentence "no number is equal to its 
>>>>>> successor" in Q is "off topic".
>>>>>>
>>>>>> Rejected, as it is a semantically valid sentence in the language 
>>>>>> of Q.
>>>>>
>>>>> If there is no finite sequence of inference steps
>>>>> between x and the axioms of Q then PTS stipulates
>>>>> that x is not semantically valid in Q.
>>>>
>>>> And since x = "no number is equal to its successor" is semantically 
>>>> valid as per the definition of Q, PTS must be discarded as useless.
>>>>
>>>
>>> So you also have no idea what Stipulative definition is.
>>>
>>> A stipulative definition is a type of definition in which
>>> a new or currently existing term is given a new specific meaning
>>>
>>> https://en.wikipedia.org/wiki/Stipulative_definition
>>
>> Not allowed, as "semantically valid" is already defined, and "no 
>> number is equal to its successor" meets that definition.
>>
> 
> Proof Theoretic Semantics (PTS) supersedes and overrules this.

Then PTS is discarded as useless because it rejects "no number is equal 
to its successor".