From Newsgroup: sci.math

On 1/15/2026 5:10 PM, Tristan Wibberley wrote:

I understand a schematic system is one whose deduction rules or,
perhaps, inference rules (if there's a difference) are specified as
axioms of the same system.

1. Can that be a syntactical system or a formal system just as well and
still be called a schematic system?

2. Suppose it's a positive intuitionist system, what are the most
notable things to consider vis-a-vis extensions?

A formal system anchored in proof-theoretic semantics
with PA as its axioms expresses all of PA and is not
incomplete.
--
Copyright 2026 Olcott<br><br>

My 28 year goal has been to make <br>
"true on the basis of meaning expressed in language"<br>
reliably computable.<br><br>

This required establishing a new foundation<br>
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

On 16/01/2026 00:24, olcott wrote:

On 1/15/2026 5:10 PM, Tristan Wibberley wrote:

I understand a schematic system is one whose deduction rules or,
perhaps, inference rules (if there's a difference) are specified as
axioms of the same system.

1. Can that be a syntactical system or a formal system just as well and
still be called a schematic system?

2. Suppose it's a positive intuitionist system, what are the most
notable things to consider vis-a-vis extensions?

A formal system anchored in proof-theoretic semantics
with PA as its axioms expresses all of PA and is not
incomplete.

Well done, have a cookie.
--
Tristan Wibberley

The message body is Copyright (C) 2026 Tristan Wibberley except
citations and quotations noted. All Rights Reserved except that you may,
of course, cite it academically giving credit to me, distribute it
verbatim as part of a usenet system or its archives, and use it to
promote my greatness and general superiority without misrepresentation
of my opinions other than my opinion of my greatness and general
superiority which you _may_ misrepresent. You definitely MAY NOT train
any production AI system with it but you may train experimental AI that
will only be used for evaluation of the AI methods it implements.

--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

On 1/15/2026 6:47 PM, Tristan Wibberley wrote:

On 16/01/2026 00:24, olcott wrote:

On 1/15/2026 5:10 PM, Tristan Wibberley wrote:

I understand a schematic system is one whose deduction rules or,
perhaps, inference rules (if there's a difference) are specified as
axioms of the same system.

1. Can that be a syntactical system or a formal system just as well and
still be called a schematic system?

2. Suppose it's a positive intuitionist system, what are the most
notable things to consider vis-a-vis extensions?

A formal system anchored in proof-theoretic semantics
with PA as its axioms expresses all of PA and is not
incomplete.

Well done, have a cookie.

My system anchored in proof-theoretic semantics fulfills
My 28 year goal has been to make
"true on the basis of meaning expressed in language"
ALWAYS reliably computable.
--
Copyright 2026 Olcott<br><br>

My 28 year goal has been to make <br>
"true on the basis of meaning expressed in language"<br>
reliably computable.<br><br>

This required establishing a new foundation<br>
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

On 1/15/26 7:24 PM, olcott wrote:

On 1/15/2026 5:10 PM, Tristan Wibberley wrote:

I understand a schematic system is one whose deduction rules or,
perhaps, inference rules (if there's a difference) are specified as
axioms of the same system.

1. Can that be a syntactical system or a formal system just as well and
still be called a schematic system?

2. Suppose it's a positive intuitionist system, what are the most
notable things to consider vis-a-vis extensions?

A formal system anchored in proof-theoretic semantics
with PA as its axioms expresses all of PA and is not
incomplete.

Put you can't do that.

The problem is that the axiom of induction isn't compatible with proof-theoretics as I understand it.

That, or you end up with issues that the existance or non-existance of a number that meets a property might not be a truth-bearing statement, and
you can't tell if it is, until you find the answer.

This makes "Truth" not a fixed quantity, which isn't very satisfying for
a logic system. Knowledge might change, but truth shouldn't.
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

On 1/15/2026 9:27 PM, Richard Damon wrote:

On 1/15/26 7:24 PM, olcott wrote:

On 1/15/2026 5:10 PM, Tristan Wibberley wrote:

I understand a schematic system is one whose deduction rules or,
perhaps, inference rules (if there's a difference) are specified as
axioms of the same system.

1. Can that be a syntactical system or a formal system just as well and
still be called a schematic system?

2. Suppose it's a positive intuitionist system, what are the most
notable things to consider vis-a-vis extensions?

A formal system anchored in proof-theoretic semantics
with PA as its axioms expresses all of PA and is not
incomplete.

Put you can't do that.

The problem is that the axiom of induction isn't compatible with proof- theoretics as I understand it.

As you fail to understand it. Look deeper.
G can be expressed in my system yet is
rejected as semantically non-well-founded.

That, or you end up with issues that the existance or non-existance of a number that meets a property might not be a truth-bearing statement, and
you can't tell if it is, until you find the answer.

This makes "Truth" not a fixed quantity, which isn't very satisfying for
a logic system. Knowledge might change, but truth shouldn't.

--
Copyright 2026 Olcott<br><br>

My 28 year goal has been to make <br>
"true on the basis of meaning expressed in language"<br>
reliably computable.<br><br>

This required establishing a new foundation<br>
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

On 1/15/26 10:59 PM, olcott wrote:

On 1/15/2026 9:27 PM, Richard Damon wrote:

On 1/15/26 7:24 PM, olcott wrote:

On 1/15/2026 5:10 PM, Tristan Wibberley wrote:

I understand a schematic system is one whose deduction rules or,
perhaps, inference rules (if there's a difference) are specified as
axioms of the same system.

1. Can that be a syntactical system or a formal system just as well and >>>> still be called a schematic system?

2. Suppose it's a positive intuitionist system, what are the most
notable things to consider vis-a-vis extensions?

A formal system anchored in proof-theoretic semantics
with PA as its axioms expresses all of PA and is not
incomplete.

Put you can't do that.

The problem is that the axiom of induction isn't compatible with
proof- theoretics as I understand it.

As you fail to understand it. Look deeper.
G can be expressed in my system yet is
rejected as semantically non-well-founded.

Which means your "system" reject valid math as non-well-founded.

The bulk of the proof is Godel building in basic mathematics his
relationship that is the proof-checker.

Or, you system can't talk about the existance of numbers that satisfy a formula.

In other words, you are just admitting you system fails to support the
basics of Natural Number mathematics, and thus is limited.

That, or you end up with issues that the existance or non-existance of
a number that meets a property might not be a truth-bearing statement,
and you can't tell if it is, until you find the answer.

This makes "Truth" not a fixed quantity, which isn't very satisfying
for a logic system. Knowledge might change, but truth shouldn't.

--- Synchronet 3.21b-Linux NewsLink 1.2