Mike, List, (I think I'm allowed one on-list response to my own topic? if not, please excuse me). As with yours, people merely need click the htlm file to see the response —which is just basic for me, in terms of potential flags and potential agreement.
Firstly, thanks Mike for going through it and I note that you're going to continue in your own way of study and research. I just wanted, on-list, to clarify some assonances and dissonances with Peirce regarding your htlm file (also, you know the lean proofing extends beyond what is in the formal PDF — that's for the final version). Apologies for how congested/dense this is, but it covers some basic points (consider infinite inquiry....). Anyway, htlm is included. Note: there are claims there I don't fully agree with (see Peirce and some of the stuff on transcendental: I begin theory-agnostic, as genuine as is possible, and so I haven't fully reconciled this with Peirce yet, and, as such, I'm wary of anything which overstates beyond what I'm certain is proven — the genuine apriori and some other interesting mathematical facts and logical results). Also, I know these ais can often misinterpret Peirce's work so I am deliberately trying to remain as far way from that (as of now) as possible until I come to Peirce explicitly —which is why input, which I've had some of already, from list-members is very useful (for next stages). Best, Jack ________________________________ From: Mike Bergman <[email protected]> Sent: Sunday, February 22, 2026 8:45 PM To: Jack Cody <[email protected]>; Peirce List <[email protected]> Cc: Edwina Taborsky <[email protected]> Subject: Re: Universal Inequivalence (using various methods). Hi Jack, Thanks for sharing this thought-provoking draft. Since I was not directly familiar with some of the background or specific terms, I asked Claude to summarize its arguments in more lay terms with specific reference to Peirce. That response is enclosed in the PDF. Also enclosed is a working dashboard to play interactively with some of these concepts. Since I don't know Lean, I asked Claude to translate it into Python and then present it as an interactive Web page. You can do the same by invoking the file in your browser. You can see the code by viewing the page source. I have noted to the list before that I work with a variety of LLMs, including ChatGPT, Grok, and Anthropic's Claude. The latter is the best for code manipulation and visualization in my opinion. I may later comment further with my own assessment. However, I thought that you and perhaps others on the list may find the enclosed of some interest. Best, Mike On 2/22/2026 12:06 PM, Jack Cody wrote: List, I've presented a rough draft of Peirce's 5.525 which, after debate, was accepted as "this is fine and logically correct" (in summary). I have been for years working on a series of papers which have to with Kant, Peirce, Godel, Tarski, et al. More... of course. I once claimed that I had proved the necessity of the ding-an-sich. I do not say this paper — rough preliminary draft — does that, but as the first in a series of such, I know that it will (what is proven, beyond trviality, is an "a priori" which fits the definition "beyond the physical". I thought it would be of interest. I've had this stress-tested a thousand times, and it's not the final version, but it is the final draft version I'll share before I proceed, with the other set of papers, which go into much more detail, prior to any serious publication. tl;dr the kind of thing required for ding-an-sich to exist (categorically) is basically proven. Whatever you want to call it (so long as you can defend it). Best, JackTitle: Peircean Interpretation â The Stratification of Access
Peircean Interpretation:
Cleaned and Annotated
What the Paper Argues â and Why Peirce Matters
The paper establishes a structural result: no system of mediated encounter can be equivalent to existence as such. This is not an epistemic claim about contingent limitations, but an a priori result about what access is. The argument uses Cantorian diagonalization, categorical morphism theory, and predicativity constraints â backed by machine-verified Lean 4 proofs.
Peircean Connection
Peirce's semiosis is irreducibly triadic: SignâObjectâInterpretant. Any attempt to collapse the triad to dyadic identity â making the sign just be the object â destroys the sign-relation. The paper's Collapse Principle formalises exactly this insight: identity of access with its object eliminates indexing, selectivity, and asymmetry, and thereby destroys the encounter-domain itself §2.3.
The Reflexive Extension Theorem gives mathematical precision to Peirce's conviction that inquiry is intrinsically open-ended: no set of signs exhausts its object; there is always a further interpretant §6.4, Lean-verified.
Alignment Table
| Peirce | Role | Paper | Status |
|---|---|---|---|
| Representamen | The mediating form â indexed, selective, asymmetric | Prime / VE(X) · Structural UI §2.1 | â Aligned |
| Object | What the sign is of â always exceeding the sign | Base (â¬) · P(O)P model §2.2 | â Aligned |
| Interpretant | Further sign generated â engine of unbounded semiosis | d* (diagonal discrimination) §6.4 | â Partial â see Flag 2 |
| Unlimited semiosis | Inquiry never reaches a final sign | ⬠> ð«, structural openness | â Partial â see Flags 1, 3 |
Flags: Where the Peircean Framing Understates the Paper's Results
"No finite set of signs exhausts its object."
The No Internal Encoding Lemma §7, Lean-verified shows there is no surjection from any set Obj onto its discrimination space (Obj â Bool), regardless of cardinality. The Reflexive Extension Theorem §6.4 constructs d* outside any fixed family DS â finite, countably infinite, or uncountably infinite.
The interpretant as "further sign generated" implies a temporal or sequential picture: one sign leads to another, and the chain never terminates.
The Reflexive Extension Theorem is not about temporal sequence. It shows that for any fixed discrimination family â considered as a completed totality â a new discrimination d* is already constructible from within S's own decoding interface. This is a structural fact about the architecture of any reflexive system, not a claim about what happens next in a process.
The Grounding of Extension Lemma §6.6 goes further: the very possibility of generating a new discrimination presupposes a ground (Base) that is not itself a product of the extension process. This ground cannot be identified with any stage or limit of the semiotic chain.
Inquiry asymptotically approaches its object. The object is a regulative ideal â approached but never reached.
The Blackbox Gap §8, Lean-verified is not an asymptotic claim. It establishes ¬(Obj â Primef) as a theorem within any adequate predicative framework. The gap is not a limit being approached; it is a structural non-equivalence that holds at every stage and cannot be closed by any extension of inquiry, however far carried.
The Surrogate Dilemma §9.1 seals this: any framework adequate to assert closure proves its own negation. The gap is not that we haven't reached the object yet â reaching it would destroy the encounter-domain (Collapse Principle, §2.3).
Corrected Overview Paragraph
Peirce's semiosis is irreducibly triadic: SignâObjectâInterpretant. Any attempt to collapse the triad to dyadic identity destroys the sign-relation. The paper's Collapse Principle formalises this: identity of access with its object eliminates the constitutive features of encounter and destroys the encounter-domain §2.3. The Reflexive Extension Theorem gives mathematical precision to the openness of semiosis: for any sign-family of any cardinality, a further discrimination is constructible from within â not merely as the next step in a sequence, but as a structural consequence of reflexivity §6.4, Lean-verified. And the Grounding of Extension Lemma shows that this openness itself presupposes a transcendental ground â Base â that is not a stage of semiosis but its condition of possibility §6.6. The object is not a regulative ideal approached asymptotically; the gap between sign and object is permanent, necessary, and proven.
Corrected Alignment Table
| Peirce | Paper | Note |
|---|---|---|
| Representamen: indexed, selective, asymmetric | Prime / Structural UI §2.1 | â Direct formal correspondence |
| Object exceeds sign | ⬠> ð«, Blackbox Gap §8 | â Proven, not merely posited |
| Further interpretant always generatable | d* constructible for any DS of any cardinality §6.4, §7 | â Stronger than Peircean: holds for all cardinalities |
| Unlimited semiosis (sequential) | Structural openness (non-sequential) | â Peirce's temporal framing understates: structural fact, not process claim |
| Object as regulative ideal | Base as transcendental ground, never a Prime-stage | â Asymptotic framing is weaker: gap is necessary, not merely unclosed |
| Self-grounding semiosis | Grounding of Extension Lemma §6.6: semiosis presupposes Base | â Peirce does not articulate the transcendental precondition the paper proves is required |
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