Solution to Carroll's Paradox

[Lewis] Carroll's Paradox (or "What the Tortoise Said to Achilles") is a logical conundrum that purports to show that modus ponens results in regressio ad infinitum. 
(references:
https://friendsoflibertylogic.blogspot.com/2008/12/carrolls-paradox.html
https://en.wikipedia.org/wiki/What_the_Tortoise_Said_to_Achilles)

Below, I try to show that if the Tortoise is claiming that his discovery that modus ponens results in regressio ad infinitum proves that modus ponens can't result in a sound conclusion, he would contradict himself.
Thanks to Hans-Hermann Hoppe!

Achilles returned to his tent and threw open the flap.

"Briseis," he shouted; "I'm home."

"Hello, dear," sang Briseis, as she tripped into the anteroom of their tent with a basin of floral water for bathing Achilles' feet. "How did your day go?"

"That cursed Tortoise," said Achilles; "He proved to me that the modus ponens must inevitably result in an infinite regression. That's the second time he's used that trick on me."
"Oh, I'm sorry dear. But how ever did he manage to do that?"
"Well, you see, he started with the premise A that 'things that are equal to the same thing are equal to each other.' Why, everyone knows this to be true. So how could I have suspected a trap? Then, he added the premise B 'the two sides of a certain triangle are equal to the same."
"Who could possibly object to that?" said Briseis.
"Well, that's what I thought," continued Achilles, "but then the Tortoise asked me if I believed that the conclusion C, 'the two sides of this certain triangle are equal to each other,' was a valid conclusion."
"Seems valid to me!" said Briseis.
"Well, of course, who wouldn't think so? But then he pointed out that I need a third premise that says that A and B lead to the conclusion C."
"No! He didn't!"
"Yes, he did. And furthermore, even when I produced such a premise, he made me produce another premise that justified the third premise and so on, ad infinitum." Achilles was near tears.
"Now, now, dear; no need to get upset. How did the Tortoise's argument end?"
"Eh? End? What do you mean 'end?' That was it---regression ad infinitum.
"But dear, was there a claim?"
"Claim? Why, of course there was. The regressio was the claim."
"But dear, did the Tortoise claim that the regression proves anything?"
"But can't you see? There was a regressio---to infinity!"
"Yes, dear, but if there was no claim, then there can be no argument. Without an argument, nothing's been proven.
"Wha-, why, pflub . . . of course something was proven. Of course it was. Why, the regressio was proven. I guess I know when something's been proven. Sure, it was." Achilles put his chin on his hand. "Wasn't it?"
"Well," said Briseis, "if something has been proven, then there must have been an argument---a conclusion based on true premises."
"Well, it may not have been stated in so many words, but there was an implicit argument."
"An implicit argument?"
"Yes, certainly."
"And how did this implicit argument run?" asked Briseis.
"Er... well, it must have been something like 'A. An infinite regression invalidates an argument. B. Modus ponens results in an infinite regression; therefore C. Arguments founded on modus ponens are invalid.'"
"I see. Well, if that was how the argument went, then might there not be a slight problem?"
"Not that I can see," said Achilles, doubtfully.
"The implied argument you gave was a modus ponens argument.
Achilles stared blankly for a moment. 
"By Zeus, so it was. That means, what; circularity?"
"Perhaps, dear, but more importantly, doesn't that constitute a pragmatic contradiction?"
"A what?"
"A pragmatic (or performative) contradiction. It means 'when the statement of an argument contradicts its claim.'"
"When the statement of an argument contradicts its . . . why, bless my myrmidons, so it does! That cursed Tortoise foiled me with a pragmatic contradiction, and I didn't even notice it! Why, why . . . that scoundrel! That blackguard . . ." Achilles leapt to his feet and dashed for the tent entrance, yelling over his should: "By all the gods and goddesses, I'll wring that impudent Tortoise's neck, and we'll have turtle soup for dinner tonight!"
"Yes, dear," said Briseis, and as Achilles ran off in search of the Tortoise, she repaired to the meat safe to see if there were any ox steaks on hand.