Abstract
In this chapter, we establish the following main result: assuming there exists a remarkable cardinal with a weakly inaccessible cardinal above it, we can force a set model of \(\mathsf{Z_3} + \mathsf{HP}\) via set forcing without the use of the reshaping technique.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
For the original version of reshaping forcing, I refer to [1, Sect. 1.3].
- 2.
We observe that Baumgartner’s proof of this fact in [3] can be done in \(\mathsf{Z_3}\) .
- 3.
For Harrington’s club shooting forcing, I refer to Sect. 4.2.
- 4.
I would like to thank W. Hugh Woodin for pointing out the problem in our original definition of B and providing this key claim.
- 5.
i.e. \(\mathbb {P}=[\omega _1]^{<\omega _1}\times [Z_{F}]^{<\omega _1}\) with \((p,q)\le (p^{\prime },q^{\prime })\) iff \(p\supseteq p^{\prime }, q\supseteq q^{\prime }\) and \(\forall \alpha \in q^{\prime }(p\cap \delta _{\alpha }\subseteq p^{\prime })\).
- 6.
i.e. If \(D\subseteq \mathbb {P}\) is a maximal antichain with \(D\in L_{\omega _2}[A_0,E]\), then \(L_{\omega _2}[A_0,E]\models |D|\le \omega _1\).
- 7.
- 8.
This fact is standard and its proof uses the standard Skolem Hull argument. We only need to check that the proof can be run in \(\mathsf{Z_3}\) which is not hard.
- 9.
The tree T is defined for definability argument. We define T to show that \(H^{\infty }\in L_{\alpha _{\eta }}[A\cap \eta ]\): we first show that \(T\in L_{\alpha _{\eta }}[A\cap \eta ]\) and then show that \(H^{\infty }\in L_{\alpha _{\eta }}[A\cap \eta ]\) via Lemma 5.10.
- 10.
To show (5.21), we use that H is continuous.
- 11.
Take a recursive function \(F:\omega \rightarrow \omega ^{\omega }\) such that \(F(\beta )(n)=f_{\beta }(n)\).
- 12.
References
Beller, A., Jensen, R.B., Welch, P.: Coding the Universe. Cambridge University Press, Cambridge (1982)
Cheng, Y.: Analysis of Martin-Harrington theorem in higher-order arithmetic. Ph.D. thesis, National University of Singapore (2012)
Baumgartner, J.: Applications of the Proper Forcing Axiom. Handbook of set-theoretic topology (Kunen, K., Vaughan, J.E.), North-Holland, Amsterdam, pp. 913–959 (1984)
Jech, T.J.: Set Theory. Third Millennium Edition, revised and expanded. Springer, Berlin (2003)
Schindler, R.: Remarkable cardinals. Infinity, Computability, and Metamathematics (Geschke et al., Eds.), Festschrift celebrating the 60th birthdays of Peter Koepke and Philip Welch, pp. 299–308
Woodin, W.H.: Personal communication with Woodin
Schindler, R.: Proper forcing and remarkable cardinals II., : Schindler, Ralf. J. Symb. Log. 66, 1481–1492 (2001)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2019 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Cheng, Y. (2019). Forcing a Model of Harrington’s Principle Without Reshaping. In: Incompleteness for Higher-Order Arithmetic. SpringerBriefs in Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-13-9949-7_5
Download citation
DOI: https://doi.org/10.1007/978-981-13-9949-7_5
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-9948-0
Online ISBN: 978-981-13-9949-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)